Improved Reall-Santos method for AdS black holes in general 4-derivative gravities

For asymptotically flat black holes, Reall-Santos method is a convenient tool to compute leading higher derivative corrections to the thermodynamic quantities without actually solving the modified field equations. However, there are subtleties in its generalization to asymptotically AdS black holes with general higher derivative corrections. First of all, it is necessary to know all the higher derivative holographic counterterms and the surface terms implementing the variational principle and subtracting the divergence. One then needs to solve for the modified AdS radius and rescale the time coordinate in an appropriate way such that the induced metric on the conformal boundary of AdS black hole is not modified. We observe that Reall-Santos method can be directly applied to a particular 4-derivative gravity model, known as the Einstein-Weyl gravity, which does not modify the AdS radius and requires only the Gibbons-Hawking-York term and holographic counterterms for the 2-derivative theory. We thus suggest that to compute the thermodynamic quantities of AdS black holes in general 4-derivative theories of gravity, one simply needs to transform it to a Einstein-Weyl gravity with identical thermodynamic variables by appropriate field redefinitions. We explicitly verify this proposal with spherically-symmetric and static charged black holes in Einstein-Maxwell theory extended with generic 4-derivative interactions.