Topological Thermodynamics Overview

• Topological thermodynamics is a framework that interprets thermodynamic states as global invariants, using concepts like winding numbers, subdivision potentials, and Euler characteristics.
• In black hole physics, critical points are classified through off-shell free energy landscapes where winding numbers diagnose phase transitions and stability.
• For quantum and nonequilibrium systems, non-extensive boundary contributions and topological indices reveal phase behavior beyond conventional local response functions.

Searching arXiv for papers on topological thermodynamics across black holes, topological matter, and nonequilibrium systems. First search: black-hole thermodynamic topology and recent classification work. Searching for Hill thermodynamics and topological phases. Searching for nonequilibrium and stochastic formulations of thermodynamic topology. Topological thermodynamics denotes a family of research programs that organize thermodynamic behavior by topological data rather than by local response functions alone. In black-hole physics, the central idea is that thermodynamic states can be treated as topological defects in an auxiliary thermodynamic parameter space; in topological quantum matter, non-extensive boundary terms in the thermodynamic potential carry the equilibrium signature of protected edge structure; in microcanonical classical-spin systems, Euler characteristics of energy sublevel sets are used to reconstruct thermodynamic observables; and in open stochastic networks, nonequilibrium drives define topological sectors of steady-state transport (Wei et al., 2024, Quelle et al., 2016, Santos et al., 2016, Mehta et al., 2022). The phrase is therefore plural in scope, and in gravitational work it must also be distinguished from the older usage in which “topological black holes” means black holes whose horizon cross sections have curvature index $k=1,0,-1$ (Zangeneh et al., 2015).

1. Meanings and conceptual domain

The modern literature uses “topological thermodynamics” in several non-equivalent but structurally related senses. One line of work treats equilibrium thermodynamic states as zeros of auxiliary vector fields, so that stability, criticality, and phase-branch structure are encoded by winding numbers and conserved topological currents. This formulation is especially developed in black-hole thermodynamics, where the off-shell free-energy landscape is mapped to a two-component field on an $(r_h,\Theta)$ or $(S,\theta)$ plane, and isolated zeros are interpreted as thermodynamic defects (Wei et al., 2021, Wei et al., 2024).

A second line of work is the Hill-thermodynamic treatment of topological quantum matter. Here the key issue is that protected edge or surface modes produce subextensive contributions to the free energy, so ordinary extensive thermodynamics is insufficient. The non-extensive subdivision potential becomes the thermodynamic carrier of boundary topology, and derivatives of that term yield measurable edge density of states, heat capacity, and boundary phase transitions (Quelle et al., 2016, Kempkes et al., 2016).

A third line is microcanonical and Morse-theoretic. In this setting, topology enters through the configuration-space manifolds $M_E=\{q:V(q)/N\le E\}$ and their critical-point structure. The Euler characteristic and the associated Euler entropy are used as topological analogs of thermodynamic state functions, with the claim that, in the thermodynamic limit and under stated hypotheses, magnetic observables and critical temperatures can be reconstructed from topology alone (Santos et al., 2016).

A fourth line concerns nonequilibrium steady states. In driven chemical networks, topological protection is reformulated in terms of chemical-potential differences, fluxes, and a drive-weighted stoichiometric matrix. In this usage, topology is not primarily a property of a Hamiltonian but of a thermodynamically biased network operator (Mehta et al., 2022).

2. Canonical constructions and topological invariants

The best-known black-hole construction starts from the generalized off-shell free energy

$\mathcal{F}=M-\frac{S}{\tau},$

where $\tau$ is an auxiliary inverse-temperature parameter. A two-component field is then introduced,

$$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$

or, in a closely related version,

$$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$

The zeros of $\phi$ are the thermodynamic states, each zero carries a winding number $w$, and the global class is

$(r_h,\Theta)$0

Within this scheme, $(r_h,\Theta)$1 corresponds to a locally stable branch and $(r_h,\Theta)$2 to a locally unstable branch; degenerate points satisfy $(r_h,\Theta)$3 and coincide with Davies points where the heat capacity diverges (Gashti et al., 2024, Wei et al., 2024).

For black-hole critical points, a related formulation begins from

$(r_h,\Theta)$4

Here a critical point is a zero of $(r_h,\Theta)$5, and the associated topological charge is

$(r_h,\Theta)$6

This yields the distinction between conventional critical points with $(r_h,\Theta)$7, novel critical points with $(r_h,\Theta)$8, and, in later work, isolated critical points with $(r_h,\Theta)$9 (Wei et al., 2021, Chu et al., 22 Oct 2025).

In Hill thermodynamics, the central invariant is not a winding number but the non-extensive subdivision potential. The energy is written as

$(S,\theta)$0

with

$(S,\theta)$1

and for ribbon geometries the grand potential is decomposed as

$(S,\theta)$2

The bulk term is extensive, while $(S,\theta)$3 is the boundary or subdivision contribution. In topological phases, $(S,\theta)$4 remains finite and its derivatives encode boundary density of states and boundary heat capacity (Quelle et al., 2016).

In the microcanonical Morse-theoretic program, the invariant is the Euler characteristic

$(S,\theta)$5

leading to the Euler entropy

$(S,\theta)$6

In nonequilibrium chemical networks, the analogous object is the drive-weighted stoichiometric matrix

$(S,\theta)$7

obtained from the nonequilibrium identity $(S,\theta)$8. In that context, topological phase transitions are controlled by gap closings of operators derived from $(S,\theta)$9 (Santos et al., 2016, Mehta et al., 2022).

3. Black-hole thermodynamic topology

The black-hole program introduced topology into criticality by assigning a topological charge to each thermodynamic critical point. In the initial classification, charged AdS black holes possess a single conventional critical point with $M_E=\{q:V(q)/N\le E\}$0, whereas charged Born–Infeld AdS black holes can exhibit both a novel critical point with $M_E=\{q:V(q)/N\le E\}$1 and a conventional one with $M_E=\{q:V(q)/N\le E\}$2. A central conclusion of that work is that only the conventional critical point can be associated with a nearby first-order phase transition; the presence of a novel critical point cannot serve as an indicator of the presence of the first-order phase transition near it (Wei et al., 2021).

The later system-wide classification of black-hole states treats black-hole solutions as topological defects in the $M_E=\{q:V(q)/N\le E\}$3 parameter space and derives four universal asymptotic classes from the behavior of the inverse Hawking temperature $M_E=\{q:V(q)/N\le E\}$4 at the minimal radius $M_E=\{q:V(q)/N\le E\}$5 and at infinity. These are

$M_E=\{q:V(q)/N\le E\}$6

with refined endpoint labels

$M_E=\{q:V(q)/N\le E\}$7

The first sign denotes the innermost branch and the second the outermost branch. The framework also fixes the low- and high-temperature limits and the alternating ordering of stable and unstable branches as $M_E=\{q:V(q)/N\le E\}$8 increases. Schwarzschild, Reissner–Nordström, Schwarzschild–AdS, and RN–AdS black holes are given as representatives of $M_E=\{q:V(q)/N\le E\}$9, $\mathcal{F}=M-\frac{S}{\tau},$0, $\mathcal{F}=M-\frac{S}{\tau},$1, and $\mathcal{F}=M-\frac{S}{\tau},$2, respectively (Wei et al., 2024).

That fourfold scheme was then extended. Static charged AdS black holes in gauged supergravity were shown to realize a new class

$\mathcal{F}=M-\frac{S}{\tau},$3

and two subclasses

$\mathcal{F}=M-\frac{S}{\tau},$4

These cases demonstrate that the same total topological number does not imply the same thermodynamic evolution: $\mathcal{F}=M-\frac{S}{\tau},$5, $\mathcal{F}=M-\frac{S}{\tau},$6, and $\mathcal{F}=M-\frac{S}{\tau},$7 all have $\mathcal{F}=M-\frac{S}{\tau},$8, yet they differ in defect-curve evolution and in their low-temperature branch content (Wu et al., 2024).

Several subsequent works turned this classification into a diagnostic of robustness and deformation sensitivity. Static multi-charge AdS black holes in gauged supergravities can undergo a novel temperature-dependent thermodynamic topological phase transition in which the total topological number changes between $\mathcal{F}=M-\frac{S}{\tau},$9 and $\tau$0 as temperature varies; the effect appears in the four-dimensional EMDA gauged supergravity black hole, the four-dimensional Horowitz–Sen black hole, and the five-dimensional Kaluza–Klein gauged supergravity black hole (Wu et al., 2024). In 4D AdS Einstein–Gauss–Bonnet black holes with Barrow, Rényi, and Sharma–Mittal entropies, bulk-boundary thermodynamics is parameter-dependent, while restricted phase space is universal and stable, always yielding $\tau$1 and $\tau$2 for all three entropy models and for all parameter choices (Gashti et al., 2024).

Concrete families illustrate both invariance and non-invariance. Kerr–Sen AdS black holes exhibit three branches with winding numbers $\tau$3 and total $\tau$4, and this topological number remains invariant under variations of the dilaton charge parameter, although the rotation parameter controls the existence of multiple critical points (Rehan et al., 25 Mar 2026). For the Bardeen black holes studied through the generalized off-shell Helmholtz free energy, regular Bardeen–AdS, Bardeen–AdS in Kiselev quintessence, Bardeen black holes in massive gravity, and Bardeen black holes in 4D Einstein–Gauss–Bonnet gravity all have $\tau$5 (Sadeghi et al., 2023). By contrast, generalized Bardeen regular black holes of the Neves–Saa family exhibit two topological defects with opposite winding numbers, so regular configurations have $\tau$6, while the Schwarzschild case contains a single unstable branch with $\tau$7 (Silva et al., 21 May 2026). Perfect fluid dark matter backgrounds also act selectively: for Schwarzschild and Kerr they do not change the thermodynamic topology, but static Hayward black holes in perfect fluid dark matter shift from $\tau$8 to $\tau$9 (Rizwan et al., 2023).

4. Topological quantum matter and equilibrium boundary thermodynamics

In topological insulators and superconductors, the central thermodynamic issue is non-extensivity. Conventional thermodynamics assumes that thermodynamic potentials scale linearly with volume, so boundary effects vanish in the thermodynamic limit. Hill thermodynamics removes that restriction by splitting the grand potential into a bulk piece and a subdivision potential. In the Bernevig–Hughes–Zhang model, the boundary term $$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$0 captures the edge density of states and the edge heat capacity, remains nonzero inside the bulk gap in the topological phase, and reveals boundary phase transitions even when they are not manifested in the bulk. The corresponding diagnostic formulas are

$$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$1

and for Dirac edge states the boundary specific heat is linear in $$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$2 (Quelle et al., 2016).

A broader Hill-thermodynamic survey established a dimensional universality of transition order. For a $$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$3-dimensional topological model, the boundary transition is of order $$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$4 and the bulk transition is of order $$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$5. The pattern was shown explicitly for the Kitaev chain and SSH model in one dimension, the Kane–Mele model in two dimensions, and the BHZ model in three dimensions. The same work also extracted finite-temperature topological phase diagrams and reported good agreement with phase boundaries obtained from the Uhlmann phase (Kempkes et al., 2016).

Higher-order topological phases require a codimension-resolved thermodynamics. For the two-dimensional quadrupolar topological insulator, the grand potential is decomposed as

$$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$6

Along the quadrupolar-to-trivial transition, the bulk contribution shows a third-order singularity, the edge contribution a second-order singularity, and the corner contribution a first-order singularity, in agreement with Josephson hyperscaling. The ordinary grand potential does not detect the trivial-to-dipolar transition, but a Wannier-band-based grand potential restores thermodynamic sensitivity to that transition (Arouca et al., 2019).

Topological Josephson junctions provide a phase-resolved version of the same idea. In a quantum spin Hall junction, the heat capacity $$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$7 develops a pronounced double peak in its phase dependence as a signature of the protected zero-energy crossing in the Andreev spectrum, and under fermion-parity conservation the free energy, Josephson current, and heat capacity become $$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$8-periodic in the superconducting phase difference (Scharf et al., 2021). Transport observables can also act as thermodynamic topology probes: in the SSH chain coupled to baths, heat current is strongly suppressed in the topological phase because non-zero-energy modes lose transmission weight at the boundaries, the curvature $$\phi=\left(\phi^{r_h},\phi^\Theta\right)=\left(\frac{\partial \mathcal{F}}{\partial r_h},-\cot\Theta\,\csc\Theta\right),$$9 is a sharp indicator of the transition, and bosonic-bath rectification is more pronounced in the topological phase for small systems (Upadhyay et al., 2023).

5. Microcanonical topology and nonequilibrium thermodynamic protection

In the microcanonical topological approach to interacting classical spins, the potential energy is treated as a Morse function and the topology of the sublevel sets $$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$0 is taken to encode equilibrium thermodynamics. For the infinite-range and short-range $$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$1 models, the Euler characteristic and Euler entropy reproduce magnetic observables such as magnetization, correlation function, susceptibility, and critical temperature. The framework emphasizes the Noncritical Neck Theorem, according to which topology does not change between critical values of the Morse function, and interprets the loss of regularity of the Morse function as the topological signature of metastable and unstable thermodynamic solutions in the mean-field case (Santos et al., 2016).

This program is explicitly asymptotic. The claim is that topology alone suffices only in the thermodynamic limit and only under the stated conditions that the critical levels lie in the same energy interval where the Boltzmann entropy is defined and that neighboring critical levels become dense as $$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$2. Within those conditions, the Euler temperature

$$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$3

reproduces $$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$4 for the infinite-range $$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$5 model and the $$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$6 behavior of the short-range chain (Santos et al., 2016).

In open stochastic chemical networks, topological protection is reformulated in purely nonequilibrium-thermodynamic terms. The concentration dynamics

$$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$7

is combined with the steady-state identity

$$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$8

to define the thermodynamic stoichiometric matrix $$\tilde{\mathcal F}=\mathcal F+\frac{1}{\sin\Theta}, \qquad \phi=\left(\frac{\partial \tilde{\mathcal F}}{\partial r_h},\frac{\partial \tilde{\mathcal F}}{\partial \Theta}\right).$$9. Gap closings of operators built from $\phi$0 produce changes of a $\phi$1 invariant, and the resulting phases exhibit robust edge-localized modes or chiral edge currents. In this formulation, topological protection is rooted in dissipation, thermodynamic driving, and entropy-producing steady states, with entropy production rate

$\phi$2

providing the thermodynamic consistency condition (Mehta et al., 2022).

6. Interpretive issues, misconceptions, and current directions

A recurrent misconception is that a single topological number completely fixes thermodynamic behavior. The later black-hole literature rejects that simplification. Systems with the same total $\phi$3 can have different low-temperature branch content and different defect-curve evolution, as shown by the distinction between $\phi$4, $\phi$5, and $\phi$6 (Wu et al., 2024). A related result is that different topological classes do not always imply different phase structures: in the Yang–Mills black hole in massive gravity, varying $\phi$7 can change the number of critical points and the topological class, while the overall phase structure remains the same (Alipour et al., 2023).

Another source of confusion concerns the meaning of “topological” in gravitational thermodynamics. In the older Einstein–dilaton and Lifshitz literature, “topological black hole” refers to the horizon geometry, with $\phi$8 spherical, $\phi$9 flat or toroidal, and $w$0 hyperbolic. That usage is thermodynamic because horizon topology affects stability and Hawking–Page-type transitions, but it is conceptually different from the newer Duan $w$1-mapping program in which topology refers to winding numbers of defects in parameter space (Zangeneh et al., 2015, Zangeneh et al., 2015, Wei et al., 2024).

The scope of topological sufficiency also remains conditional. The microcanonical Euler-entropy program is presented as reliable for the exactly soluble spin models studied, but not as a theorem for all systems; the paper itself notes counterexamples and open questions involving noncompact configuration spaces and missing hypotheses in the underlying topological theorem (Santos et al., 2016). In black-hole thermodynamics, framework choice matters as well: for AdS Einstein–Gauss–Bonnet black holes with non-extensive entropies, bulk-boundary thermodynamics is parameter-sensitive, whereas restricted phase space is topologically invariant with $w$2 and $w$3 under all conditions considered (Gashti et al., 2024).

A further current direction is the relation between thermodynamic topology and dynamics. Quasinormal-mode calculations near critical points indicate that $w$4 cases in RN–AdS, Born–Infeld–AdS, and quantum anomalous black holes display very similar dynamical characteristics, while $w$5 and $w$6 do not yet show a comparably uniform pattern. This suggests a nontrivial connection between topological thermodynamics and black-hole dynamics, but the present evidence is still class-specific rather than fully universal (Chu et al., 22 Oct 2025).

Taken together, these developments indicate that topological thermodynamics is not a single formalism but a research domain unified by one methodological ambition: to recast thermodynamic phase structure, stability, and criticality in terms of global invariants. Depending on the system, those invariants are winding numbers of off-shell defects, subdivision potentials of boundary modes, Euler characteristics of energy landscapes, or drive-weighted topological indices of nonequilibrium networks.