Black hole thermodynamics at 4 derivatives, natural variables and BPS limits

We study Einstein-Maxwell theory in $D \geq 3$ spacetime dimensions including all Lorentz-invariant parity-even four-derivative couplings. Building on the results of arXiv:2312.11610, we consider static, charged, asymptotically flat black hole solutions to first order in the higher-derivative expansion. In $D=4$ and $D=5$, we compute the corrected black hole thermodynamics and compare with the Reall-Santos prescription based on the two-derivative background, highlighting a subtlety when both inner and outer horizons are involved. By introducing natural variables, as in arXiv:2304.07320, we recast the on-shell actions in terms of left- and right-moving chemical potentials, which significantly simplifies the analysis. We also compute first-order thermodynamic corrections for the most general rotating black holes in $D=4$ and $D=5$, without modifying the background solutions. We identify a novel BPS-like limit in $D=4$, extending known supergravity results beyond their traditional domain of validity. Finally, in $D=5$, the analysis of BPS and almost BPS limits enables an independent verification of the five-dimensional BPS thermodynamics. We clarify the origin of a discrepancy in the literature concerning higher-derivative supergravity localization, sharpening the tension between direct computations and predictions based on the $D=4$/$D=5$ connection.