Thermodynamics of Hairy Black Holes in Quantum Regimes: Insights from Horndeski Theory

Abstract: We study non-perturbative quantum gravitational corrections to the thermodynamics and quantum work distribution of the $n$-dimensional Schwarzschild--Tangherlini--Anti-de Sitter black hole. Starting from the corrected entropy $S = S_0 + η\, e^{-S_0}$, where $S_0$ is the Bekenstein--Hawking entropy, we derive the modified specific heat, internal energy, Helmholtz free energy, and Gibbs free energy in closed form. The specific heat retains the classical divergence at $r_h^{*}=l\sqrt{(n-3)/(n-1)}$ for $n\geq 4$, but the quantum correction suppresses its magnitude by up to $78\%$ at small horizon radii. In the extended phase space, the uncharged black hole admits no van der Waals critical point; however, the non-perturbative correction induces a Hawking--Page transition for $n\geq 4$ that is absent in the semi-classical limit. The corrected Gibbs free energy turns negative at small $r_h$, opening a thermodynamic channel with no classical counterpart. Using the Jarzynski equality and Jensen inequality, we obtain the quantum work distribution during evaporation. The free energy difference $ΔF$ between two black hole states undergoes a sign reversal at small horizon radii for $n\geq 4$ when $η=1$, flipping the average quantum work from negative to positive. This sign reversal grows with the spacetime dimension, reaching $\langle W\rangle \approx +4.31$ for $n=10$. These findings demonstrate that non-perturbative quantum gravitational effects qualitatively alter the phase structure and evaporation energetics of AdS black holes, and they cannot be captured by perturbative corrections alone.

• The paper demonstrates that non-perturbative quantum corrections significantly modify black hole thermodynamics, unveiling dimension-dependent phase transitions.
• The methodology uses corrected entropy with exponential terms and the Jarzynski equality to rigorously quantify quantum work distributions.
• The findings provide a new framework for understanding black hole evaporation and holographic duality in high-dimensional AdS spacetimes.

Thermodynamics and Quantum Work Distribution of Hairy Black Holes in Horndeski Theory with Non-Perturbative Corrections

Introduction

This paper provides a comprehensive study of non-perturbative quantum gravitational corrections to the thermodynamics and quantum work distribution of $n$-dimensional Schwarzschild–Tangherlini–AdS (ST–AdS) black holes, with a focus on the quantum regime where classical (semi-classical) thermodynamics breaks down. Starting from the corrected entropy $S = S_0 + \eta e^{-S_0}$, where $S_0$ is the Bekenstein–Hawking entropy and $\eta$ controls the strength of the exponential correction, the authors derive closed-form expressions for the modified specific heat, internal energy, Helmholtz free energy, and Gibbs free energy. The analysis is performed for arbitrary spacetime dimension $n$, uncovering a distinctive dimension-dependent structure of quantum effects. The interplay between gravitational thermodynamics, quantum corrections, and non-equilibrium quantum work is addressed, especially through the Jarzynski equality applied to black hole evaporation scenarios. A key result is the demonstration that non-perturbative quantum corrections produce qualitative alterations in the thermodynamic phase structure and energetics, especially in the high-dimensional small-horizon regime.

Quantum Corrections to Black Hole Thermodynamics

The standard semi-classical picture of black hole (BH) thermodynamics, epitomized by the Bekenstein–Hawking entropy–area relation, is known to be modified by quantum gravitational corrections. While perturbative corrections yield logarithmic terms, recent developments argue for the universality of non-perturbative exponential corrections of the form $e^{-S_0}$, especially in the Planck regime and for quantum-sized BHs. The paper emphasizes that these corrections do not significantly affect macro BHs, but are non-negligible in the quantum domain ($S_0 \sim 1$).

The underlying geometry is the $n$-dimensional ST–AdS metric, which generalizes Schwarzschild–AdS solutions to arbitrary dimensions. The event horizon $r_h$ and the Hawking temperature $T$ are dimension-dependent, leading to distinct thermodynamic features as $n$ increases. The quantum-corrected entropy

$$S = \frac{\omega}{2} r_h^{n-2} + \eta e^{-\frac{\omega}{2} r_h^{n-2}}$$

(where $\omega$ depends on $n$, the hypersphere volume) is shown to interpolate smoothly from classical to quantum regime, acting as a regulator preventing the entropy from vanishing for $r_h \rightarrow 0$.

Figure 2: Corrected entropy $S$ as a function of $r_h$ for $n=4$ and $l=1$; exponential corrections dominate at small $r_h$.

For small $r_h$, the quantum correction can exceed $1000\%$ (relative to $S_0$), while for $r_h > 1$ it becomes rapidly negligible.

Phase Structure and Thermodynamic Stability

The specific heat $C_V$, derived from the corrected entropy, retains the classical divergence at $r_h^* = l\sqrt{(n-3)/(n-1)}$ for $n \geq 4$, which marks a second-order phase transition boundary between stable and unstable thermodynamic branches. Quantum corrections suppress the magnitude of $C_V$ in the small-$r_h$ regime by up to $78\%$ for $n=4$ and $l=1$. The suppression is controlled by the factor $1 - \eta e^{-S_0}$, which becomes strongly relevant for quantum-sized BHs, effectively reducing thermal response near the Planck scale. For $n=3$, no divergence occurs and $C_V$ remains positive for all $r_h$; thus, three-dimensional AdS BHs show robust stability.

The extended thermodynamic phase space—treating the cosmological constant as a pressure—reveals no van der Waals-like critical points for uncharged ST–AdS black holes. The equation of state reduces to a special case of the van der Waals equation, and all nontrivial features in the small-horizon regime are quantum-driven rather than arising from classical or perturbative effects.

Quantum-Induced Phase Transitions

An important result is the identification of a Hawking–Page-type phase transition produced by quantum corrections. In the uncorrected ($\eta=0$) theory, for $n \geq 4$ the Gibbs free energy $G$ remains positive for all $r_h$, precluding a phase transition. The exponential correction in the entropy, however, introduces a new channel at small $r_h$ where $G$ crosses zero and becomes negative—a behavior absent in the semi-classical regime.

This quantum-induced Hawking–Page transition persists for all $n \geq 4$, with the transition temperature $T_{\mathrm{HP}}$ and critical horizon radius $r_h^{\mathrm{HP}}$ increasing monotonically with $n$ (e.g., $T_{\mathrm{HP}} \approx 0.446$ for $n=4$, $T_{\mathrm{HP}} \approx 1.371$ for $n=10$ with $l=1$). For $n=3$, the transition exists even without quantum corrections, aligning with the classical result.

Internal Energy and Free Energies

The corrected internal energy is derived in terms of incomplete gamma functions and generalized Laguerre polynomials. The non-perturbative correction raises the internal energy of small BHs by factors exceeding order unity, but remains negligible for $r_h \gg 1$. The Helmholtz and Gibbs free energies incorporate quantum corrections that generate the aforementioned phase transitions and new stability regimes.

For large BHs, quantum corrections are suppressed exponentially, but in the quantum regime, the modified thermodynamics opens up channels with negative free energy absent in classical treatments.

Quantum Work Distribution via Jarzynski Equality

The authors utilize the Jarzynski equality, a central result in non-equilibrium statistical mechanics, to relate the quantum-corrected free energy difference $\Delta F$ between initial and final BH states (labelled by $r_{h1}$, $r_{h2}$) to the exponential average of quantum work performed during evaporation:

$$\langle \exp(-\beta W) \rangle = \exp(\beta \Delta F).$$

The analysis demonstrates that in the quantum regime ($r_{h2} \ll 1$), $\Delta F$ undergoes a sign reversal for $n \geq 4$, meaning that the average quantum work switches from negative (work required) to positive (work extracted). This effect is dimension-enhanced, with $\langle W \rangle$ reaching values as high as $+4.31$ for $n=10$ and small final horizons. This is in stark contrast to the uncorrected or perturbatively corrected theories, where $\Delta F$ remains non-negative.

(Figure 2)

Figure 3: Corrected entropy $S$ as a function of $r_{h}$ showing the quantum “floor” at small radius induced by non-perturbative corrections.

The statistical weights of final microstates in evaporation, determined by $e^{-\Delta F}$, can be exponentially enhanced in higher dimensions, fundamentally altering the expected end states of quantum black hole evaporation.

Theoretical and Practical Implications

The findings imply that non-perturbative quantum corrections can induce novel thermodynamic behavior, including new phase transitions and a restructuring of the energy and work landscape for black holes in the quantum regime. This is especially relevant for AdS/CFT applications, quantum black hole evaporation dynamics, and microstate counting.

These results also provide a basis for future studies of charged, rotating, or more general “hairy” black holes, where the interplay between quantum corrections and classical critical phenomena (e.g., vdW-like transitions, critical exponents) can be systematically explored. In the AdS/CFT context, the dual CFT partition function must incorporate these corrections, prompting further investigation into the CFT duals of quantum-induced transitions.

Conclusion

This work establishes rigorous, closed-form expressions for non-perturbative quantum gravitational corrections to the thermodynamics and quantum work distribution of $n$-dimensional ST–AdS black holes. The key results include the quantum suppression and sign reversal effects in specific heat and quantum work, and the emergence of quantum-induced Hawking–Page transitions in high dimensions. The exponential sensitivity of these corrections, especially in the small-horizon, high-dimensional regime, signals the necessity of non-perturbative treatments when considering quantum black hole evaporation and phase structure. These conclusions open pathways for deeper exploration of quantum gravity signatures in black hole physics and their ramifications for holographic field theories, non-equilibrium quantum thermodynamics, and the statistical interpretation of black hole microphysics.