More Exact Thermodynamic Analysis of Topological Black Holes in $R^2$ Gravity

TL;DR

This work analyzes how small statistical fluctuations modify the thermodynamics of topological black holes in $R^2$ gravity. By deriving a logarithmic entropy correction $S=S_0-\alpha \ln S_0$ with $S_0=\frac{96\pi r_+^{2}}{l^{2}}$ and $\alpha=\tfrac{1}{2}$, the authors compute corrected thermodynamic quantities $E$, $F$, $V$, $P$, $H$, and $G$ from the first-law relations and study stability via the corrected specific heat $C_v$. The results show that internal energy diverges for small horizon radii, the corrected Gibbs free energy becomes large and positive at small $r_+$, and a double phase transition in stability emerges due to fluctuations, especially for small black holes. These findings highlight a substantial fluctuation-induced restructuring of the thermodynamics and stability landscape for topological black holes in quadratic gravity, with potential implications for holography and quantum gravity contexts.

Abstract

This study investigates the thermodynamics of topological black hole solutions in $R^{2}$ gravity, incorporating the effects of small statistical fluctuations up to first-order corrections. We precisely calculate entropy, internal energy, Helmholtz free energy, specific heat, enthalpy, and Gibbs free energy, accounting for perturbative thermal corrections. Our results reveal that the internal energy of small black holes diverges asymptotically due to these fluctuations. The corrected Gibbs free energy attains asymptotically high values for small horizon radii. In contrast, the equilibrium Gibbs free energy approaches zero. Additionally, we assess the stability of the black hole in the presence of these thermal fluctuations. We find that, in contrast to the equilibrium state, the thermal fluctuation introduces a double phase transition to the stability of the black hole. Our analysis reveals that the influence of fluctuations is notably significant, primarily for small black holes. These findings offer new insights into the thermodynamic properties of topological black holes in the presence of thermal fluctuations.

Figures (7)

• Figure 1: Entropy versus Horizon radius. The red line denotes $\alpha =0$, and the black line denotes $\alpha=0.5$. Here, $l=1$.
• Figure 2: Internal energy versus horizon radius. Here, blue line denotes $\alpha=0$, and red line denotes $\alpha=0.5$. Here, we set $l=1$, $k=1$.
• Figure 3: The Helmholtz free energy versus horizon radius. The blue line denotes $\alpha=0$, and the red line denotes $\alpha=0.5$. Here, we set $l = 1$, $k = 1$.
• Figure 4: Pressure versus horizon radius. $k = 1$and $l = 1$.
• Figure 5: Enthalpy versus horizon radius. Here, blue line denotes $\alpha=0$ and red line denotes to $\frac{1}{2}$. Here, we set $k= 1$, $l =1$.