Comments on Sen's Classical Entropy Function for Static and Rotating AdS$_4$ Black Holes

We consider various aspects of Sen's classical entropy function formalism for asymptotically AdS$_4$ black holes with emphasis on its efficacy to capture higher derivative corrections to the Bekenstein-Hawking entropy. The formalism has the important advantage of being based on near-horizon symmetries and does not require knowledge of the full interpolating supergravity solution, nor of its AdS$_4$ asymptotics. For the static case, we focus on applying the entropy function formalism in the presence of various higher derivative terms motivated in conformal supergravities; we find agreement with recently reported results utilizing the full black hole solutions and Wald's entropy formula. For the rotating case, we demonstrate that a modified version of the formalism generates a background that coincides precisely with the Bardeen-Horowitz limit of known rotating, electrically charged AdS$_4$ black holes and provides a swift approach to the black hole entropy, including higher derivatives corrections. We conclude that Sen's classical entropy function formalism is a viable and highly efficient approach to capturing higher-derivative corrections to the entropy of asymptotically AdS$_4$ black holes albeit naturally missing certain relations arising from global aspects of the full black hole solution.