Thermodynamics of Einstein-Gauss-Bonnet Black Holes and Ensemble-averaged Theory

In this paper, using the ensemble-averaged theory, we define the thermodynamic free energy of Einstein-Gauss-Bonnet (EGB) black holes in anti-de Sitter (AdS) spacetime. This approach derives the gravitational partition function by incorporating non-saddle geometries besides the classical solutions. Unlike the sharp transition points seen in free energy calculated via saddle-point approximation, the ensemble-averaged free energy plotted against temperature shows a smoother behavior, suggesting that black hole phase transitions may be viewed as a small $G_N$ (Newton's gravitational constant) limit of the ensemble theory. This is similar to the behavior of black hole solutions in Einstein's gravity theory in AdS spacetime. We have obtained an expression for the quantum-corrected free energy for EGB-AdS black holes, and in the six-dimensional case, we observe a well-defined local minimum after the transition temperature which was absent in the earlier analysis of the classical free energy landscape. Furthermore, we expand the ensemble-averaged free energy in powers of $G_N$ to identify non-classical contributions. Our findings indicate that the similarities in the thermodynamic behavior between five-dimensional EGB-AdS and Reissner-Nordstr\"om-AdS (RN-AdS) black holes, as well as between six-dimensional EGB-AdS and Schwarzschild-AdS black holes, extend beyond the classical regime.