Off-shell phase diagram of BPS black holes in AdS$_5$

We construct the off-shell free energy of supersymmetric black holes in AdS$_5$, and study the phase diagram in various limiting cases, with particular emphasis on BPS thermodynamics. The changes to the free energy following from the four-derivative corrections to five-dimensional minimal gauged supergravity action are computed, and the modifications to the phase diagram are studied. Starting from Landau's theory, an exact method is systematically developed to construct the off shell BPS free energy, which in certain limiting cases, can be rearranged in terms of an effective energy and entropy of the system, with the later being conjugate to an effective BPS temperature. The off-shell BPS phase diagram shows features which resemble the phases of general AdS Schwarzschild black holes, with some nuances in the asymptotic structure, modified by four-derivative corrections. Using AdS/CFT, phenomenological effective potentials in the boundary gauge theory are proposed, dual to both general black holes and their BPS counterparts. The saddle points of the effective potential capture the various locally stable and unstable phases of the gauge theory at finite temperature and chemical potential.