First order corrections to black hole thermodynamics: a simple approach enhanced

The inclusion of higher derivative terms in the gravitational action brings about corrections to the original forms of black hole solutions and thermodynamics. A simple and valuable approach has emerged over two decades ago, which states that the first order corrections to black hole thermodynamics, caused by higher derivative terms, can be achieved without explicitly solving the perturbed metric. However, a direct application of this approach may potentially yield incorrect answers as it comes to AdS black holes, which poses a challenge to the valuable approach. In this paper, we identify the subtlety associated with AdS black holes is that certain higher derivative terms may alter the asymptotic structure of the spacetime. By taking this subtlety into account, we successfully enhance the original approach and resolve the mentioned problem. Furthermore, we make an interesting observation that the extremalization analysis that underlies this approach also plays a fundamental role in the perturbation theories of quantum mechanics and other disciplines.