Extended Thermodynamical Topology

• Extended Thermodynamical Topology is a framework that encodes black-hole phase structure through integer invariants derived from vector field zeros on thermodynamic parameter space.
• It reinterprets the cosmological constant as pressure and the black-hole mass as enthalpy, enabling analysis via generalized free energies and equations of state.
• A hierarchical sequence of topological charges unifies phase branches, spinodal curves, and critical points to distinguish conventional from novel thermodynamic behaviors.

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Extended thermodynamical topology is a topological framework for black-hole thermodynamics in which the phase structure of a gravitational system is encoded in integer invariants constructed from vector fields on thermodynamic parameter space. In the extended thermodynamic picture, the cosmological constant is promoted to a pressure, $P=-\Lambda/(8\pi G)$, the black-hole mass is interpreted as enthalpy, and criticality is analyzed through equations of state and generalized free energies. Within this setting, zeros of suitably defined vector fields represent on-shell black-hole states or higher-order degeneracies, while winding numbers, Brouwer degrees, and related charges classify stable and unstable branches, spinodal curves, and critical points (Wei et al., 2021, Wu et al., 3 Aug 2025).

1. Thermodynamic setting and scope

The thermodynamic substrate of the subject is the extended phase space of AdS black holes. In this formulation one identifies the negative cosmological constant with thermodynamic pressure,

$P=-\frac{\Lambda}{8\pi G},$

interprets the black-hole mass $M$ as enthalpy $H$, and writes the first law as

$dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$

with $V=(\partial M/\partial P)_S$ the thermodynamic volume and $x^i,Y_i$ additional conjugate pairs such as charge–potential or angular momentum–angular velocity. One may equivalently work with an equation of state $P=P(V,T,\ldots)$, or with $T=T(S,P,\ldots)$, and define critical points by the usual inflection-point conditions in the $(P,V)$ or $P=-\frac{\Lambda}{8\pi G},$0 planes (Wei et al., 2021).

A later usage of the term “extended thermodynamical topology” broadens the meaning of the subject. Rather than restricting topology to the classification of equilibrium branches alone, it unifies three previously distinct topological numbers associated with phases, spinodal curves, and critical points into a hierarchy of $P=-\frac{\Lambda}{8\pi G},$1th-order topological charges $P=-\frac{\Lambda}{8\pi G},$2. In that unified formulation, $P=-\frac{\Lambda}{8\pi G},$3 characterizes phase branches, $P=-\frac{\Lambda}{8\pi G},$4 locates Davies-type spinodal points, $P=-\frac{\Lambda}{8\pi G},$5 classifies critical points, and higher $P=-\frac{\Lambda}{8\pi G},$6 detect more degenerate mergers of thermodynamic branches (Wu et al., 3 Aug 2025).

The literature therefore uses the phrase in two closely related senses: first, topology applied to black holes in extended thermodynamic phase space; second, a hierarchical topological classification of black-hole phase structure that includes but is not limited to that phase space. This suggests that the field has evolved from a method for diagnosing isolated critical points into a broader organizational scheme for black-hole thermodynamics.

2. Core mathematical construction

The original black-hole implementation follows Duan’s topological current $P=-\frac{\Lambda}{8\pi G},$7-mapping theory. One chooses two coordinates $P=-\frac{\Lambda}{8\pi G},$8 in thermodynamic parameter space; in the first application, $P=-\frac{\Lambda}{8\pi G},$9 and $M$0, where $M$1 is an auxiliary angular variable chosen so that the zero of the vector field lies at $M$2. Starting from $M$3, one eliminates $M$4 through $M$5, defines

$M$6

and sets

$M$7

Critical points of the thermodynamic system are then exactly the zeros $M$8 (Wei et al., 2021).

The corresponding topological current is built from the normalized field $M$9: $H$0 Using the Jacobian $H$1, one obtains

$H$2

so the current localizes at zeros of $H$3. The total charge on a two-dimensional region $H$4 is

$H$5

where $H$6 is the Hopf index, $H$7 the Brouwer degree, and $H$8 the winding number of the $H$9-th zero. For an isolated zero with linear expansion $dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$0, the local winding number reduces to

$dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$1

A negative determinant gives $dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$2, and a positive determinant gives $dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$3 (Wei et al., 2021).

A complementary formulation uses the generalized off-shell Helmholtz free energy

$dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$4

where $dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$5 is the Euclidean time period. One then defines the vector field on the $dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$6 plane,

$dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$7

so that zeros satisfy $dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$8 and the on-shell condition $dM=T\,dS+V\,dP+\sum_i Y_i\,dx^i,$9. In this off-shell language, the total topological charge is

$V=(\partial M/\partial P)_S$0

with $V=(\partial M/\partial P)_S$1 the winding number around each zero (Sadeghi et al., 2023).

A later reformulation dispenses with the auxiliary angle altogether. In that approach one defines an off-shell grand free energy $V=(\partial M/\partial P)_S$2 by Legendre-transforming the ADM mass with respect to all exchangeable quantities and constructs the gradient-flow vector field $V=(\partial M/\partial P)_S$3. The resulting conserved topological tensor or higher-dimensional topological current yields a charge equal to the sum of the indices of all zeros. In this construction, the topological charge is independent of the ensemble parameters and is argued to depend only on the asymptotic geometry (Nam, 28 Oct 2025).

3. Hierarchy of topological invariants

The unified framework introduces a sequence of vector fields $V=(\partial M/\partial P)_S$4 whose zeros encode progressively more degenerate thermodynamic structures. The zeroth-order object is the off-shell free-energy field

$V=(\partial M/\partial P)_S$5

whose zeros satisfy $V=(\partial M/\partial P)_S$6, equivalently $V=(\partial M/\partial P)_S$7. Its local Jacobian is

$V=(\partial M/\partial P)_S$8

so the local winding number satisfies

$V=(\partial M/\partial P)_S$9

with $x^i,Y_i$0 for locally stable branches and $x^i,Y_i$1 for unstable ones (Wu et al., 3 Aug 2025).

Imposing $x^i,Y_i$2 and examining the remaining degeneracy leads to the first-order field

$x^i,Y_i$3

Its zeros locate Davies-type spinodal points, i.e. divergence points of the heat capacity. The Jacobian becomes

$x^i,Y_i$4

hence

$x^i,Y_i$5

which records the change of stability across the spinodal (Wu et al., 3 Aug 2025).

At the next stage, imposing $x^i,Y_i$6 yields

$x^i,Y_i$7

Its zeros satisfy

$x^i,Y_i$8

which is the standard criticality condition in this framework. The associated Jacobian is

$x^i,Y_i$9

so

$P=P(V,T,\ldots)$0

and the sign distinguishes “conventional” from “novel” critical points (Wu et al., 3 Aug 2025).

The full recursive construction is

$P=P(V,T,\ldots)$1

with simultaneous zero conditions

$P=P(V,T,\ldots)$2

The global invariant

$P=P(V,T,\ldots)$3

depends only on the asymptotic behavior of $P=P(V,T,\ldots)$4, and higher nonzero $P=P(V,T,\ldots)$5 diagnose more degenerate coalescences of thermodynamic branches (Wu et al., 3 Aug 2025).

This hierarchy is connected to critical exponents. If $P=P(V,T,\ldots)$6 has a simple zero at $P=P(V,T,\ldots)$7, then

$P=P(V,T,\ldots)$8

Under a perturbation of a control parameter $P=P(V,T,\ldots)$9, one finds

$T=T(S,P,\ldots)$0

The framework therefore associates higher-order zeros with modified critical exponents, and explicitly notes that even $T=T(S,P,\ldots)$1 gives novel critical exponents; Widom is obeyed, while Rushbrooke is generally violated for $T=T(S,P,\ldots)$2 (Wu et al., 3 Aug 2025).

4. Representative black-hole classifications

The first explicit application concerned the topology of black-hole thermodynamics at critical points. For the charged AdS black hole with fixed charge $T=T(S,P,\ldots)$3, the Hawking temperature is

$T=T(S,P,\ldots)$4

with one critical solution at $T=T(S,P,\ldots)$5, $T=T(S,P,\ldots)$6. The Jacobian has negative sign, so $T=T(S,P,\ldots)$7, and the total topological charge is $T=T(S,P,\ldots)$8. For the Born–Infeld AdS black hole, by contrast, there are two zeros: $T=T(S,P,\ldots)$9 with $(P,V)$0 and $(P,V)$1 with $(P,V)$2, giving net charge $(P,V)$3. Only $(P,V)$4 supports a small–large first-order transition (Wei et al., 2021).

This distinction led to the terminology of conventional and novel critical points. A critical point with $(P,V)$5 is “conventional”: the usual small–large black-hole first-order coexistence curve emanates from it in the $(P,V)$6–$(P,V)$7 plane. A critical point with $(P,V)$8 is “novel”: it is mathematically a critical point of $(P,V)$9, but no first-order line actually extends from it, because Maxwell’s area law fails. One of the central corrections introduced by the topological viewpoint is therefore that the mere existence of a critical point in $P=-\frac{\Lambda}{8\pi G},$00 does not guarantee a physical first-order transition (Wei et al., 2021).

Subsequent analyses enlarged the catalogue of examples:

System	Local pattern	Global classification
Charged AdS	one zero with $P=-\frac{\Lambda}{8\pi G},$01	$P=-\frac{\Lambda}{8\pi G},$02
Born–Infeld AdS	$P=-\frac{\Lambda}{8\pi G},$03	$P=-\frac{\Lambda}{8\pi G},$04
AdS RN and AdS EGB in BBT	$P=-\frac{\Lambda}{8\pi G},$05	$P=-\frac{\Lambda}{8\pi G},$06
AdS EPYM in BBT, RPS, EPST	$P=-\frac{\Lambda}{8\pi G},$07	$P=-\frac{\Lambda}{8\pi G},$08
Dyonic AdS with quasitopological electromagnetism	one, three, or five states with alternating signs	$P=-\frac{\Lambda}{8\pi G},$09

For AdS Reissner–Nordström and AdS Einstein–Gauss–Bonnet black holes in the bulk–boundary framework, the three zeros carry $P=-\frac{\Lambda}{8\pi G},$10, so $P=-\frac{\Lambda}{8\pi G},$11; in restricted phase space each has a single zero with $P=-\frac{\Lambda}{8\pi G},$12, again $P=-\frac{\Lambda}{8\pi G},$13. AdS Einstein–power–Yang–Mills black holes instead have two zeros $P=-\frac{\Lambda}{8\pi G},$14 and total $P=-\frac{\Lambda}{8\pi G},$15, and the same pattern persists in bulk–boundary, restricted, and extended phase space thermodynamics (Sadeghi et al., 2023).

Dyonic AdS black holes with quasitopological electromagnetism provide a distinct phenomenon: multiple defect curves. Depending on the region of the $P=-\frac{\Lambda}{8\pi G},$16 diagram, there may be one, three, or five black-hole states; the corresponding winding numbers alternate as $P=-\frac{\Lambda}{8\pi G},$17, so the algebraic sum remains

$P=-\frac{\Lambda}{8\pi G},$18

even in the presence of two separated first-order coexistence curves (Chen et al., 2024).

The quantum BTZ black hole extends the framework in another direction. In a $P=-\frac{\Lambda}{8\pi G},$19-domain description based on

$P=-\frac{\Lambda}{8\pi G},$20

one finds a global winding number $P=-\frac{\Lambda}{8\pi G},$21, but the local interpretation is branch-dependent because $P=-\frac{\Lambda}{8\pi G},$22 changes sign. Passing to an entropy-domain construction with two foliations produces a topological number directly linked to stability, with $P=-\frac{\Lambda}{8\pi G},$23 and $P=-\frac{\Lambda}{8\pi G},$24. In this formulation a topological transition occurs in the “cold” branch: $P=-\frac{\Lambda}{8\pi G},$25 The same work explicitly argues for topological numbers beyond the conventional values of $P=-\frac{\Lambda}{8\pi G},$26 (Wu et al., 2024).

5. Ensemble equivalence, dimensions, and couplings

A major theme of the subject is whether thermodynamic topology depends on the chosen ensemble or thermodynamic representation. One approach uses the residue method instead of a vector-field winding number. There one defines a characteristic function $P=-\frac{\Lambda}{8\pi G},$27, such as $P=-\frac{\Lambda}{8\pi G},$28, whose poles coincide with phase-transition points, and assigns the local charge

$P=-\frac{\Lambda}{8\pi G},$29

Applied to charged Reissner–Nordström–AdS black holes, this method yields

$P=-\frac{\Lambda}{8\pi G},$30

in extended thermodynamics, mixed thermodynamics, and boundary CFT thermodynamics alike. This is formulated as a theorem of bulk–boundary topological equivalence (Zhang et al., 2023).

The bulk–boundary, restricted phase space, and extended phase space comparison has also been carried out within the off-shell Helmholtz framework. For AdS EPYM black holes, the local topological numbers and total charge are the same in all three descriptions,

$P=-\frac{\Lambda}{8\pi G},$31

and the nonlinear Yang–Mills charge parameter $P=-\frac{\Lambda}{8\pi G},$32 shifts the location of the zeros without changing their number or sign pattern. This suggests an ensemble-robust global classification, even though local branch structure can differ between frameworks for other black holes such as AdS RN and AdS EGB (Sadeghi et al., 2023).

The role of couplings and spacetime dimension has been explored in several later families. In ultraspinning Kerr–AdS black holes in arbitrary dimension, only two thermodynamic topological structures appear: the standard class $P=-\frac{\Lambda}{8\pi G},$33 and the subclass $P=-\frac{\Lambda}{8\pi G},$34. The global topological number is always $P=-\frac{\Lambda}{8\pi G},$35; even-dimensional cases are always $P=-\frac{\Lambda}{8\pi G},$36, while odd-dimensional black holes with maximal rotations realize $P=-\frac{\Lambda}{8\pi G},$37. No further classes or subclasses appear (Tian et al., 5 Feb 2026).

Einstein–Maxwell–Dilaton theories enlarge the classification further. In dimensions $P=-\frac{\Lambda}{8\pi G},$38, varying the dilaton coupling $P=-\frac{\Lambda}{8\pi G},$39 near the critical value

$P=-\frac{\Lambda}{8\pi G},$40

induces transitions between distinct thermodynamic topological phases. In particular, these models exhibit a new class denoted

$P=-\frac{\Lambda}{8\pi G},$41

characterized by pair creation of $P=-\frac{\Lambda}{8\pi G},$42 defects at a generation temperature and annihilation of the unstable branch at a higher annihilation temperature, leaving a single stable branch and net $P=-\frac{\Lambda}{8\pi G},$43. The Gubser–Rocha model belongs to this classification (Babaei-Aghbolagh et al., 20 Aug 2025).

Regular black holes generated purely by higher-curvature gravity provide a different universality statement. For the infinite-tower models studied in $P=-\frac{\Lambda}{8\pi G},$44, the Hawking temperature has at least one zero at small horizon radius, and the off-shell topological construction yields exactly one stable small branch and one unstable large branch with local pattern $P=-\frac{\Lambda}{8\pi G},$45. The global charge is therefore always $P=-\frac{\Lambda}{8\pi G},$46, described in refined notation as the class $P=-\frac{\Lambda}{8\pi G},$47, and this is proposed as a topological invariant of the construction method itself (Wang et al., 2024).

6. Interpretation, misconceptions, and broader context

Several recurrent misconceptions are corrected by the literature. The first is that any mathematical critical point in $P=-\frac{\Lambda}{8\pi G},$48 marks the endpoint of a physical first-order line. The distinction between conventional and novel critical points shows this is false: a critical point with $P=-\frac{\Lambda}{8\pi G},$49 can exist without any first-order coexistence curve, because Maxwell’s area law fails there (Wei et al., 2021).

A second misconception is that equal total topological charge implies identical local thermodynamic structure. This is also false. AdS RN and AdS EGB can have $P=-\frac{\Lambda}{8\pi G},$50 in both bulk–boundary and restricted phase space while exhibiting different numbers of local zeros in those frameworks; likewise, nonextensive entropy deformations of AdS RN black holes can change the number of zeros while preserving $P=-\frac{\Lambda}{8\pi G},$51. In the bulk–boundary framework with Rényi entropy, $P=-\frac{\Lambda}{8\pi G},$52 gives a single zero with $P=-\frac{\Lambda}{8\pi G},$53, whereas the $P=-\frac{\Lambda}{8\pi G},$54 Bekenstein–Hawking limit gives three zeros $P=-\frac{\Lambda}{8\pi G},$55; in restricted phase space, however, one again finds a single $P=-\frac{\Lambda}{8\pi G},$56 zero for all $P=-\frac{\Lambda}{8\pi G},$57, and similarly for Sharma–Mittal entropy (Gashti, 2024).

A third issue concerns the role of auxiliary variables. Much of the established formalism uses $P=-\frac{\Lambda}{8\pi G},$58 or $P=-\frac{\Lambda}{8\pi G},$59 to anchor the second component of the vector field. A newer construction removes this nonphysical variable and instead works directly with gradients of the off-shell grand potential in the true thermodynamic exchange variables. In that setting, the topological charge is independent of the ensemble parameters, black holes with the same background geometry share the same charge, Minkowski and de Sitter black holes have $P=-\frac{\Lambda}{8\pi G},$60, and AdS black holes have $P=-\frac{\Lambda}{8\pi G},$61 (Nam, 28 Oct 2025). This suggests a transition from a two-dimensional defect picture toward a higher-dimensional index theorem.

The broader context is the extended-thermodynamics program on which the topology is imposed. That background includes, among other examples, Gauss–Bonnet–AdS black holes where $P=-\frac{\Lambda}{8\pi G},$62 is treated as a pressure and can yield one or two critical points depending on dimension, topology, and charge (Xu et al., 2013); topological charged black holes in massive gravity’s rainbow, where van der Waals-like behavior is observed for $P=-\frac{\Lambda}{8\pi G},$63 (Hendi et al., 2016); and Taub–NUT/Bolt–AdS spaces, where the discrete curvature parameter $P=-\frac{\Lambda}{8\pi G},$64 qualitatively alters the phase diagram and the existence of stable windows (Lee, 2015). These constructions do not by themselves constitute thermodynamic topology, but they define the phase structures whose global organization the topological framework classifies.

Within this landscape, extended thermodynamical topology has become a precise vocabulary for the integer classification of black-hole thermodynamics. Its defining objects are generalized free energies, vector-field zeros, winding numbers, and higher-order charges; its central achievements are the separation of genuine and nongenuine criticality, the organization of spinodal and critical structures into a single hierarchy, and the demonstration that in many cases the total topological class is robust under changes of ensemble, dimension, and microscopic couplings (Wu et al., 3 Aug 2025). The open directions explicitly suggested in the literature include rotating AdS black holes, multi-charge solutions, higher-derivative gravities, non-gravitational fluids, higher-form charges, dynamical cosmological constant, ensemble-dependence in holography, and a dual CFT interpretation of the index (Wei et al., 2021, Nam, 28 Oct 2025).