Thermodynamic Topology of Black Holes

• Thermodynamic topology is a framework that represents thermodynamic states and critical points as topological defects in vector fields derived from thermodynamic potentials.
• It employs methods like Duan’s φ-mapping theory and gradient-flow formulations to assign integer winding numbers, linking local stability to phase behavior.
• The approach classifies systems into conventional and novel critical points or stable and unstable branches, offering insights into phase transitions across diverse ensembles and modified gravities.

Searching arXiv for papers on thermodynamic topology to ground the article in the current literature. arXiv search query: thermodynamic topology black hole topology wound number free energy Thermodynamic topology is a research program in which thermodynamic states, critical points, or equilibrium black-hole branches are represented as topological defects of auxiliary vector fields constructed from thermodynamic potentials. In the black-hole literature, two main formulations recur. One treats thermodynamic critical points as zeros of a vector field built from a temperature-based potential and assigns them topological charges through Duan’s $\phi$-mapping theory; this yields a classification into conventional and novel critical points (Wei et al., 2021). The other, now widely used, treats black-hole branches themselves as zeros of a vector field derived from an off-shell free energy, with local winding numbers $w_i$ and total topological charge $W=\sum_i w_i$ providing a global classification of the thermodynamic landscape (Gogoi et al., 2023). Later developments extend this framework across ensembles, modified gravities, non-extensive entropies, holographic bulk-boundary correspondences, and gradient-flow formulations that remove the auxiliary angular variable and instead use the off-shell grand free energy in physical thermodynamic state space (Nam, 28 Oct 2025).

1. Historical emergence and conceptual scope

Thermodynamic topology entered the black-hole literature as a topological reformulation of thermodynamic criticality. In the first explicit construction, a critical point was identified not only by local differential conditions such as

$$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$

but also as a zero of an auxiliary vector field on a thermodynamic parameter space (Wei et al., 2021). In that framework, each critical point carries an integer-valued topological charge, and critical points fall into two classes: conventional critical points with charge $-1$ and novel critical points with charge $+1$ (Wei et al., 2021). A conventional critical point is the one from which the familiar first-order small/large black-hole coexistence line can emanate, whereas a novel critical point does not by itself indicate such a nearby first-order transition (Wei et al., 2021).

A second strand of the subject reinterprets black-hole equilibrium branches as thermodynamic defects in an auxiliary space built from the horizon radius and an angular variable. The central object is an off-shell free energy, and the zeros of the associated vector field correspond to on-shell black-hole states (Gogoi et al., 2023). Each zero carries a local winding number, and the total topological number

$W=\sum_i w_i$

is used as a global classification invariant (Ladghami et al., 7 Jul 2025). In this literature, positive winding numbers are associated with thermodynamically stable branches, and negative winding numbers with unstable branches (Sadeghi et al., 2024).

The scope of thermodynamic topology has expanded rapidly. It has been applied to charged, dyonic, phantom, Horava–Lifshitz, BTZ, Taub–NUT-like, brane-world, Kiselev, and ultraspinning Kerr–AdS black holes; to modified-gravity settings such as $F(R)$ gravity, massive gravity, Einstein–bumblebee gravity, and gravity’s rainbow; and to generalized entropy frameworks including Barrow entropy and Tsallis statistics (Panah et al., 2024, Chen et al., 2024, Hazarika et al., 2023, Wu et al., 2024, Azreg-Aïnou et al., 22 Sep 2025, Zafar et al., 1 Mar 2026, Ladghami et al., 7 Jul 2025). Related work has also proposed residue-based topological charges for bulk and boundary thermodynamics in AdS/CFT (Zhang et al., 2023).

2. Mathematical formulations

Two constructions dominate the literature.

The first begins from a temperature function $T(S,P,x^i)$. One eliminates a thermodynamic variable, typically $P$, using the extremality condition

$w_i$0

and defines

$w_i$1

A two-component vector field is then introduced,

$w_i$2

so that critical points are the common zeros of $w_i$3 and $w_i$4 (Wei et al., 2021). Because $w_i$5 enforces $w_i$6, the remaining zero condition reproduces the thermodynamic criticality condition. In this setting, thermodynamic critical points become topological defects.

The second begins from a generalized off-shell free energy, usually written as

$w_i$7

or, in ensembles with fixed potentials, from the corresponding Legendre-transformed form (Gogoi et al., 2023, Sadeghi et al., 2024). The standard auxiliary-space vector field is

$w_i$8

or equivalent variants such as

$w_i$9

and, in one no-boundary cylinder construction,

$W=\sum_i w_i$0

with $W=\sum_i w_i$1 (Azreg-Aïnou et al., 22 Sep 2025). The zeros satisfy $W=\sum_i w_i$2 or, in the cylinder construction, $W=\sum_i w_i$3, together with the on-shell relation $W=\sum_i w_i$4 (Azreg-Aïnou et al., 22 Sep 2025, Tian et al., 5 Feb 2026).

A later reformulation eliminates the nonphysical angular variable entirely. In that approach, one uses the off-shell grand free energy

$W=\sum_i w_i$5

for charged black holes, or its $W=\sum_i w_i$6-variable generalization,

$W=\sum_i w_i$7

and defines the gradient-flow vector field directly in physical thermodynamic state space,

$W=\sum_i w_i$8

Its components are

$W=\sum_i w_i$9

so the zeros enforce both thermal and chemical equilibrium (Nam, 28 Oct 2025). This formulation is presented as avoiding the earlier auxiliary $$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$0-construction and as tying thermodynamic topology to a physical ensemble in which black holes exchange both energy and matter with the environment (Nam, 28 Oct 2025).

3. Topological invariants, currents, and stability interpretation

Most of the literature uses Duan’s $$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$1-mapping theory. For a normalized field

$$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$2

the topological current is

$$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$3

in the two-component construction (Ladghami et al., 7 Jul 2025, Tian et al., 5 Feb 2026). It can be rewritten as

$$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$4

so the current is supported only at zeros of the vector field (Sadeghi et al., 2024). The total topological charge is

$$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$5

where $$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$6 is the Hopf index, $$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$7 the Brouwer degree, and $$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$8 the local winding number (Ladghami et al., 7 Jul 2025, Zafar et al., 1 Mar 2026).

In many applications, the sign of the local winding number is interpreted as a local stability diagnostic: $$\left(\frac{\partial T}{\partial S}\right)_{P,x^i}=0,\qquad \left(\frac{\partial^2 T}{\partial S^2}\right)_{P,x^i}=0,$$9 denotes a stable branch and $-1$0 an unstable branch (Zafar et al., 1 Mar 2026, Sadeghi et al., 2024). This interpretation often agrees with the sign of the specific heat (Sadeghi et al., 2024), and some papers state it explicitly as a rule (Tian et al., 5 Feb 2026). In the gradient-flow grand-free-energy formulation, the Jacobian is related to the heat capacity at fixed chemical potential,

$-1$1

so the local index matches the sign of $-1$2 in the cases considered (Nam, 28 Oct 2025).

The relation between topology and stability is not always uniform. In the quantum BTZ case, the standard parameter-based vector field yields a topological number $-1$3, but quantum corrections render entropy non-monotonic in the parameter $-1$4, and the correspondence between winding sign and stability flips when $-1$5 changes sign (Wu et al., 2024). Rebuilding the topology with entropy as the domain variable restores a globally meaningful stability interpretation and reveals a topological transition $-1$6 at $-1$7 (Wu et al., 2024). This suggests that the physical interpretation of the invariant can depend on the thermodynamic coordinate used in the construction.

4. Ensembles and thermodynamic potentials

Ensemble dependence is a central theme. In dyonic AdS black holes, the topology differs across canonical, mixed, and grand canonical ensembles (Gogoi et al., 2023). In that study, canonical and mixed ensembles each possess one conventional critical point with charge $-1$8, while the grand canonical ensemble has no critical point (Gogoi et al., 2023). In the defect picture, canonical and mixed ensembles both have total topological charge $-1$9, whereas the grand canonical ensemble has $+1$0 or $+1$1 depending on the electric and magnetic potentials (Gogoi et al., 2023).

A closely related pattern appears in phantom AdS black holes in massive gravity. In the canonical ensemble, there is a conventional critical point $+1$2 with topological charge $+1$3, but only in the ordinary Einstein–Maxwell case $+1$4; in the grand canonical ensemble, no critical point is found (Chen et al., 2024). In the defect analysis, the canonical ensemble always has total topological charge $+1$5, while the grand canonical ensemble has $+1$6 or $+1$7 depending on the electric potential (Chen et al., 2024).

The importance of ensemble choice extends beyond critical points. In topological black holes in $+1$8-ModMax gravity’s rainbow, the fixed-$+1$9 and fixed-$W=\sum_i w_i$0 ensembles yield different classes $W=\sum_i w_i$1, with the class controlled by the horizon topology $W=\sum_i w_i$2, the sign of the scalar curvature $W=\sum_i w_i$3, and, in one fixed-$W=\sum_i w_i$4 regime, the rainbow function $W=\sum_i w_i$5 (Panah et al., 2024). In Horava–Lifshitz black holes, a fixed-$W=\sum_i w_i$6 ensemble and a fixed-$W=\sum_i w_i$7 ensemble likewise lead to distinct topological classifications across spherical, flat, and hyperbolic horizons (Hazarika et al., 2023).

These examples show that thermodynamic topology is generally not an invariant of the metric alone. In much of the defect literature, it is an invariant of a chosen thermodynamic description: the thermodynamic potential, Legendre frame, and boundary conditions all matter (Gogoi et al., 2023, Chen et al., 2024).

5. Global classifications and recurrent charge patterns

A recurrent theme is the emergence of simple integer classes. Many papers adopt $W=\sum_i w_i$8, $W=\sum_i w_i$9, and $F(R)$0 as the primary global categories.

In the Tsallis-statistics framework, four-dimensional Schwarzschild and uncharged higher-dimensional black holes are classified into three classes: $F(R)$1 interpreted respectively as stable, critical, and unstable (Ladghami et al., 7 Jul 2025). For Schwarzschild,

$F(R)$2

where $F(R)$3 is the non-extensive parameter in $F(R)$4 (Ladghami et al., 7 Jul 2025). Charged black holes in the same framework exhibit only two classes, $F(R)$5 and $F(R)$6, with the latter corresponding to coexistence of one stable and one unstable branch (Ladghami et al., 7 Jul 2025).

In brane-world black holes with Barrow entropy, the global charge is either $F(R)$7 or $F(R)$8, and the dominant control parameter is the dark matter parameter $F(R)$9, while the Barrow deformation parameter $T(S,P,x^i)$0 mainly reshapes the local thermodynamic structure without generally changing $T(S,P,x^i)$1 (Zafar et al., 1 Mar 2026). In quantum-corrected AdS-Reissner–Nordström black holes in Kiselev spacetime, most parameter choices give $T(S,P,x^i)$2, but special cases realize $T(S,P,x^i)$3 and $T(S,P,x^i)$4 (Sadeghi et al., 2024).

Some papers refine the classification further. Ultraspinning Kerr–AdS black holes in arbitrary dimension realize only two thermodynamic topological structures,

$T(S,P,x^i)$5

with both having total $T(S,P,x^i)$6 but differing in endpoint branch organization: $T(S,P,x^i)$7 has stable innermost and outermost branches, while $T(S,P,x^i)$8 has an unstable innermost branch and a stable outermost branch (Tian et al., 5 Feb 2026). The distinct subclass $T(S,P,x^i)$9 appears only for odd-dimensional black holes with the maximal number of independent rotations (Tian et al., 5 Feb 2026).

A more recent development argues that topological charge may be an invariant of asymptotic spacetime background. In the grand-free-energy gradient-flow framework, asymptotically Minkowski-flat and de Sitter spacetimes are assigned $P$0, while AdS spacetimes have $P$1 (Nam, 28 Oct 2025). This suggests a different style of universality, tied not to ensemble choice but to background geometry.

6. Phase structure, branch creation, and topological transitions

Thermodynamic topology is often used to re-express familiar phase structure in geometric language. Generation points and annihilation points are where pairs of defect branches are created or destroyed, typically determined by the simultaneous conditions

$P$2

in the defect formalism (Gogoi et al., 2023). Because the created or annihilated pair carries opposite winding numbers, the total topological charge is conserved across the event (Chen et al., 2024).

In canonical and mixed dyonic AdS ensembles, below critical pressure there are three branches with winding numbers $P$3, corresponding to stable small, unstable intermediate, and stable large black holes; above critical pressure only one stable branch remains, but the total charge stays $P$4 (Gogoi et al., 2023). Similar branch structures appear in phantom AdS black holes in massive gravity (Chen et al., 2024) and in the fixed-$P$5 ensemble of topological black holes in $P$6-ModMax gravity’s rainbow (Panah et al., 2024).

Thermodynamic topology has also been used to characterize continuous phase-transition lines. In four- and five-dimensional Horava–Lifshitz black holes with spherical horizons, the thermodynamics exhibits a $P$7-line, a line of second-order phase transitions akin to superfluid $P$8 (Hazarika et al., 2023). In the fixed-$P$9 ensemble, the small/intermediate/large branch pattern appears for $w_i$00, but the total topological charge remains $w_i$01 across the $w_i$02-transition and changes only at $w_i$03, where it becomes $w_i$04 (Hazarika et al., 2023). This separates local branch reorganization from global topological class.

In the quantum BTZ case, the topological transition is of a different kind: the entropy-based total topological number changes from $w_i$05 to $w_i$06 at $w_i$07, reflecting the migration of the cold black-hole state between branches with different stability (Wu et al., 2024). This is not a standard van der Waals-like first-order transition; it is a branch/stability transition controlled by the entropy-turning point $w_i$08 (Wu et al., 2024).

7. Extensions beyond standard black-hole chemistry

Several lines of work broaden thermodynamic topology beyond standard Einstein–Maxwell–AdS thermodynamics.

One line replaces the usual Bekenstein–Hawking entropy. In Tsallis statistics, the entropy is

$w_i$09

and the non-extensive parameter $w_i$10 changes the temperature scaling, heat capacity, and topological class (Ladghami et al., 7 Jul 2025). In brane-world black holes with Barrow entropy,

$w_i$11

the deformation parameter $w_i$12 changes local heat-capacity zeros and divergences, while the global charge is governed mainly by the dark matter parameter $w_i$13 (Zafar et al., 1 Mar 2026).

Another line focuses on Lorentz-violating or modified-gravity systems. In Einstein–bumblebee gravity, a Taub–NUT-like black hole has an off-shell free-energy landscape with a single maximum and total topological charge

$w_i$14

The Lorentz-breaking parameter rescales $w_i$15, $w_i$16, and $w_i$17 by $w_i$18, but does not change the topological class because it shifts only the location, not the number or index, of the critical points (Azreg-Aïnou et al., 22 Sep 2025). In $w_i$19-ModMax gravity’s rainbow, the topological class depends on the ensemble, curvature sign, and horizon topology, demonstrating sensitivity to modified gravity, nonlinear electrodynamics, and energy-dependent geometry (Panah et al., 2024). In phantom AdS black holes in massive gravity, the existence of a conventional critical point is restricted to the classical Einstein–Maxwell sector $w_i$20 in the canonical ensemble (Chen et al., 2024).

A third line explores holographic and bulk-boundary formulations. In charged AdS black holes and their dual CFTs, a residue-based method assigns topological charges to phase transitions rather than to equilibrium branches (Zhang et al., 2023). In that framework, canonical criticality carries charge $w_i$21, canonical first-order coexistence carries charge $w_i$22, and Hawking–Page or Hawking–Page-like transitions in the grand canonical ensemble carry charge $w_i$23, with the same value in extended bulk thermodynamics, mixed thermodynamics, and boundary CFT thermodynamics (Zhang et al., 2023). This suggests that topological thermodynamic data can be holographically preserved (Zhang et al., 2023).

8. Interpretive issues and methodological debates

Several methodological issues recur.

A first concerns the auxiliary angle $w_i$24 or $w_i$25. Much of the literature uses the vector field

$w_i$26

or close variants (Gogoi et al., 2023, Sadeghi et al., 2024, Tian et al., 5 Feb 2026). This construction is mathematically convenient, but the angular variable is nonphysical. A newer formulation argues that the need for $w_i$27 indicates that earlier treatments omitted matter exchange with the environment and that once chemical equilibrium is included, the correct object is the gradient of an off-shell grand free energy in physical thermodynamic state space (Nam, 28 Oct 2025). This proposal reframes thermodynamic topology as a topology of actual thermodynamic variables rather than an auxiliary extension.

A second issue concerns the meaning of the invariant. In much of the defect literature, $w_i$28 is interpreted as a global classification of branch structure, often close to “stable minus unstable branches” (Ladghami et al., 7 Jul 2025). But the quantum BTZ analysis shows that this interpretation can fail when entropy is not monotonic in the parameter used to build the vector field (Wu et al., 2024). That work argues that entropy, or a monotonic function of it, is the physically natural domain variable and that multibranched systems may require a foliation-by-foliation construction (Wu et al., 2024).

A third issue concerns rigor and formal unity. Some papers employ the full vector-field/winding-number machinery with explicit zero-point analysis (Sadeghi et al., 2024, Ladghami et al., 7 Jul 2025), while others use different objects. For example, the study of quantum RN black holes introduces contour-integral “topological numbers” based on derivatives of free energy and entropy,

$w_i$29

but does not define a normalized thermodynamic vector field, Brouwer degree, or winding-number map in the standard sense (Chen, 2024). That paper itself notes that the entropy-based number appears more physically meaningful than the free-energy-based one and that the ensemble structure is not rigorously specified (Chen, 2024). This suggests that “thermodynamic topology” is not yet a single universally standardized formalism.

9. Broader significance

The common motivation across these works is that topology offers a global classifier of thermodynamic structure. Local observables such as temperature curves, heat capacities, and response functions diagnose criticality and stability branch by branch. Thermodynamic topology instead packages this information into integer-valued invariants attached either to critical points, phase-transition loci, or equilibrium-branch defects (Wei et al., 2021, Ladghami et al., 7 Jul 2025, Zhang et al., 2023).

This suggests several broader implications. First, topological data can distinguish thermodynamic systems with similar equations of state but different branch organization, as in charged AdS versus Born–Infeld AdS black holes (Wei et al., 2021). Second, topological classifications can expose which deformations are merely quantitative and which are qualitative. Lorentz symmetry breaking in Einstein–bumblebee gravity changes thermodynamic quantities but not the topological class (Azreg-Aïnou et al., 22 Sep 2025), whereas quantum correction plus Kiselev matter can change the total charge from $w_i$30 to $w_i$31 or $w_i$32 (Sadeghi et al., 2024). Third, some authors argue that topological charge may encode universal information tied to asymptotic geometry itself (Nam, 28 Oct 2025).

A plausible implication is that thermodynamic topology may become a comparative language across black-hole families, ensembles, and even holographic dual descriptions. The bulk-boundary residue method already presents phase-transition charges that match exactly between black-hole thermodynamics and boundary CFT thermodynamics (Zhang et al., 2023). Another plausible implication is that as more multibranched or quantum-corrected systems are analyzed, topological numbers beyond the common values $w_i$33 may become relevant, especially in foliated or multi-defect settings (Wu et al., 2024).

Thermodynamic topology therefore designates not a single invariant or a single construction, but a family of closely related methods for encoding thermodynamic criticality, branch structure, and stability into topological objects—vector-field zeros, winding numbers, defect currents, residue signs, and index sums. Across the present literature, the subject is technically diverse but conceptually unified by one idea: black-hole thermodynamics can be organized by topological structures that persist under deformations more robustly than ordinary local response data alone (Wei et al., 2021, Ladghami et al., 7 Jul 2025, Nam, 28 Oct 2025).