Extremalization approach to black hole thermodynamics: perturbations around higher-derivative gravities

When higher-derivative terms are added to a gravitational action, black hole solutions and their thermodynamic properties are generally corrected. Recent progress has shown that, by treating higher-derivative operators as perturbations, the first-order corrections to black hole thermodynamics can be obtained without explicit knowledge of the corresponding perturbed black hole solutions. This result can be understood as a consequence of an extremalization principle underlying the Euclidean action formulation of black hole thermodynamics. In this paper, we emphasize that this extremalization approach is not restricted to perturbations around Einstein gravity. Instead, it can be applied to perturbations of more general higher-derivative gravity theories whose black hole solutions are already known and can be taken as the zeroth-order background. As an explicit illustration, we consider Einstein–Gauss–Bonnet gravity as the zeroth-order theory and study the first-order thermodynamic corrections induced by further higher-order curvature operators. We show that these corrections can be derived without solving the perturbed black hole solutions, both in asymptotically flat and asymptotically AdS spacetimes.