Hawking-Page Phase Transition

• The Hawking-Page phase transition is a first-order gravitational change between black holes and thermal backgrounds in AdS/dS spacetimes, pivotal in black-hole thermodynamics.
• It employs detailed methods—such as Gibbs free energy analysis and universal Ruppeiner invariants—to quantify critical temperature and latent heat effects.
• Research spans various frameworks including higher-curvature gravity and holographic dualities, offering insights into quantum gravity and gauge theory correspondences.

The Hawking-Page phase transition is a prototypical first-order gravitational phase transition between black holes and thermal backgrounds, primarily studied in anti-de Sitter (AdS), de Sitter (dS), and extended frameworks. It plays a central role in black-hole thermodynamics, holographic dualities, quantum gravity, and the statistical mechanics of gravitational systems. The following presents a comprehensive review of the Hawking-Page transition, including its mathematical structure, universality, kinetic and stochastic features, topological classification, and its interpretation in gauge/gravity dualities.

1. Fundamental Structure and Thermodynamics

The classic Hawking-Page transition occurs in asymptotically AdS spacetimes, where the gravitational partition function is dominated by two saddle points: a thermal AdS background and a large black hole. For a four-dimensional Schwarzschild-AdS black hole with line element

$$ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 d\Omega^2,\quad f(r) = 1 - \frac{2M}{r} + \frac{r^2}{L^2},$$

the Hawking temperature and Gibbs free energy read

$$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$

with $r_+$ the horizon radius.

At low temperature, the globally stable phase is thermal AdS. At high temperature, the black hole solution dominates. The transition temperature $T_{HP}$ is determined by $G_{BH}(T_{HP}) = G_{thermal\, AdS}$, yielding

$T_{HP} = \frac{1}{\pi L}$

for neutral AdS black holes (Belhaj et al., 2020).

Generically, the transition is first-order: the entropy and internal energy jump discontinuously, and there is latent heat. The free energy profile exhibits a local minimum (large black hole), an unstable maximum (small black hole), and a global minimum (thermal AdS phase at low $T$) (Li et al., 2021).

2. Universality, Dualities, and Ruppeiner Invariants

A critical feature of the Hawking-Page transition is the presence of universal dimensionless constants quantifying the relative change in entropy and temperature: $$C_S = \frac{S_{HP} - S_{min}}{S_{min}}, \quad C_T = \frac{T_{HP} - T_{min}}{T_{min}},$$ where $S_{min}, T_{min}$ are the entropy and temperature at the minimal point of the $T(S)$ curve, below which no black hole can exist. For four-dimensional AdS-Schwarzschild,

$$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$0

These values persist for a broad class of non-rotating and charged black holes, and they have closed-form generalizations in higher dimensions

$$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$1

(Belhaj et al., 2020).

There exists a precise duality: $$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$2 suggesting a holographic structural mapping between $$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$3-dimensional boundary and $$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$4-dimensional bulk temperatures. At the Hawking-Page point, the normalized Ruppeiner scalar curvature (a measure of thermodynamic microstructure interactions) is universal: $$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$5 identifying an interaction threshold necessary for black-hole dominance (Wei et al., 2020).

3. Generalizations: Other Geometries, Matter, and Higher-Curvature Gravity

The transition structure is robust across several modifications:

• BTZ black holes and 3D gravity: The Hawking-Page transition can be formulated between the BTZ black hole ($$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$6) and the thermal AdS soliton ($$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$7). There exists a mass gap, and off-shell free energies are constructed by carefully accounting for conical singularities. The transition is realized as tunneling across the off-shell free-energy barrier, with a critical temperature $$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$8 (Eune et al., 2013).
• dS Spacetimes with Multiple Horizons: In de Sitter backgrounds (e.g., four-dimensional dS with nonlinear charge corrections), the HP coexistence curve in the $$T(r_+) = \frac{f'(r_+)}{4\pi} = \frac{1}{4\pi r_+}\left(1 + \frac{3r_+^2}{L^2}\right), \quad G(T) = M(r_+) - T S(r_+),$$9 plane forms a closed loop bounded from above, exhibiting two branches associated with different horizon separation regimes—the upper branch corresponding to well-separated horizons, and the lower to tightly coupled ones. This is markedly different from the AdS case where $r_+$0 is unbounded from above (Du et al., 2021).
• Black holes with boundaries/cavities: Placing an asymptotically flat Schwarzschild or Reissner–Nordström black hole in a reflecting cavity restores equilibrium and yields an HP-like transition. The transition temperature $r_+$1 increases with cavity “pressure” $r_+$2 and decreases with the electric potential $r_+$3. There exists an upper bound on $r_+$4 for the existence of the transition (Zhao et al., 2020).
• Higher-derivative and Gauss–Bonnet corrections: Four-dimensional Einstein–Gauss–Bonnet (4EGB) gravity exhibits HP transitions only in a finite range of pressure $r_+$5, with $r_+$6 suppressed monotonically as the Gauss–Bonnet coupling $r_+$7 increases. The entropy receives a logarithmic correction, and additional features such as suppressed $r_+$8 and pressure bounds appear (Wang et al., 2020, Su et al., 2019, Camanho et al., 2012).
• Cosmological analogues and McVittie black holes: A Hawking-Page–like first-order phase transition can occur between a cosmological FRW spacetime and the McVittie black hole, with the cosmic fluid work density acting as the effective pressure. The critical temperature is $r_+$9 (Abdusattar et al., 2022).

4. Kinetics, Nonequilibrium, and Stochastic Effects

The Hawking-Page transition can be studied beyond equilibrium by mapping the off-shell Gibbs/Bragg-Williams free energy onto a classical reaction-diffusion or Fokker-Planck problem. Key elements are:

• Treating the horizon radius ($T_{HP}$0) as an order parameter evolving under stochastic fluctuations and dissipation.
• The mean first-passage time (MFPT) for transition is governed asymptotically (for additive noise and overdamped limit) by a Kramers-type formula: $T_{HP}$1 where $T_{HP}$2 is the barrier height (Li et al., 2021).
• Including black-hole evaporation as a reaction term shifts the transition kinetics, permitting transitions even when thermal activation is suppressed.
• Non-Markovian friction or multiplicative noise (noise amplitude dependent on $T_{HP}$3) introduces complex dynamical crossovers: stronger small-radius noise can postpone or inhibit black-hole nucleation, shifting $T_{HP}$4. Memory effects can either promote or hinder barrier crossing depending on correlation time and friction kernel structure, as quantified by the Grote–Hynes rate theory (Li et al., 2022, Ho, 29 Sep 2025).

5. Topological Classification and Universality

Recent work applies topological methods to the free-energy landscape—especially when including non-extensive (Rényi) statistics for black holes in asymptotically flat space. Three critical mappings are used to diagnose phase structure:

• The $T_{HP}$5-mapping singles out global Hawking–Page-type transitions (temperature independent), assigning a unit winding number ($T_{HP}$6) at the HP point.
• The $T_{HP}$7-mapping detects temperature-dependent Van der Waals–type transitions.
• The $T_{HP}$8-mapping assigns topological charges to equilibrium phases—stable (large BH, $T_{HP}$9), unstable (small BH, $G_{BH}(T_{HP}) = G_{thermal\, AdS}$0), and inflection points ($G_{BH}(T_{HP}) = G_{thermal\, AdS}$1).

In the off-shell Bragg–Williams landscape, these vortex/anti-vortex features are regulation- and ensemble-independent. The topological invariants agree for asymptotically flat (Rényi statistics) and AdS (Gibbs–Boltzmann statistics) black holes, indicating that the two classes of systems fall into the same topological universality classes for both HP and Van der Waals-like transitions (Barzi et al., 2023).

Mapping	Transition Type	Topological Charge
$G_{BH}(T_{HP}) = G_{thermal\, AdS}$2	Hawking-Page	$G_{BH}(T_{HP}) = G_{thermal\, AdS}$3
$G_{BH}(T_{HP}) = G_{thermal\, AdS}$4	Van der Waals	model/ensemble dep.
$G_{BH}(T_{HP}) = G_{thermal\, AdS}$5	Equilibrium phases	$G_{BH}(T_{HP}) = G_{thermal\, AdS}$6

This formalism not only clarifies black-hole equilibrium and transition structure but also hints at a correspondence between the thermodynamic topology of black holes in asymptotically flat and AdS spacetimes.

6. Information Release and Holography

In the context of gauge/gravity duality, the Hawking-Page transition is dual to a confinement–deconfinement transition in the dual field theory (e.g., $G_{BH}(T_{HP}) = G_{thermal\, AdS}$7 SYM in 4D, QCD in AdS/QCD models). At the critical temperature, the bulk transition from thermal AdS to the AdS black hole corresponds to a sharp jump in the CFT free energy.

Including fine-grained entropy with island contributions in the bulk, the Hawking-Page transition is tied to a discontinuous drop in the Page curve, and all black hole information is released at $G_{BH}(T_{HP}) = G_{thermal\, AdS}$8 in the sense of a first-order information jump. This universality is expected to extend to other black-hole first-order phase transitions (e.g., small/large or hairy/neutral) and thus offers an alternative, abrupt "quench"-like mechanism for information release, distinct from standard slow Hawking evaporation (Sun, 2021).

7. Extensions, Modifications, and Open Directions

• Maxwell construction in $G_{BH}(T_{HP}) = G_{thermal\, AdS}$9: Applying the equal-area principle in $T_{HP} = \frac{1}{\pi L}$0 "flattens" the negative-heat-capacity region and may shift the HP transition temperature down to $T_{HP} = \frac{1}{\pi L}$1, removing metastability (Spallucci et al., 2013).
• Generalized topologies: On products of spheres $T_{HP} = \frac{1}{\pi L}$2, generalized HP transitions arise between bulk phases of different topologies, with the order of the transition changing for total dimension $T_{HP} = \frac{1}{\pi L}$3 (Aharony et al., 2019).
• Higher-derivative/bubble nucleation: In higher-curvature gravities such as Gauss–Bonnet, novel HP transitions can be mediated by bubble nucleation, involving two spacetimes with different effective cosmological constants (Camanho et al., 2012).
• AdS/QCD and density effects: At finite baryon chemical potential $T_{HP} = \frac{1}{\pi L}$4, the HP critical temperature decreases, matching the qualitative features of the QCD phase diagram, and terminates at a finite density below which the system is always deconfined (Braga et al., 2024).
• Massive gravity and non-compact boundaries: In massive AdS gravity, an HP transition can occur even with a planar (non-compact) boundary, with the graviton mass introducing an infrared scale analogous to a confinement scale (Braga et al., 16 Sep 2025).

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In summary, the Hawking-Page transition is a structurally robust, quantitatively universal, and topologically invariant feature of black-hole thermodynamics across diverse gravitational frameworks. It has deep implications for quantum gravity, thermodynamic stability, black-hole information transfer, and the phase structure of dual gauge theories, and is accessible to both equilibrium and nonequilibrium/topological methods (Belhaj et al., 2020, Eune et al., 2013, Barzi et al., 2023, Li et al., 2021, Wei et al., 2020).