  We present a holographic derivation of the entropy of supersymmetric
asymptotically AdS$_5$ black holes. We define a BPS limit of black hole
thermodynamics by first focussing on a supersymmetric family of complexified
solutions and then reaching extremality. We show that in this limit the black
hole entropy is the Legendre transform of the on-shell gravitational action
with respect to three chemical potentials subject to a constraint. This
constraint follows from supersymmetry and regularity in the Euclidean bulk
geometry. Further, we calculate, using localization, the exact partition
function of the dual $\mathcal{N}=1$ SCFT on a twisted $S^1\times S^3$ with
complexified chemical potentials obeying this constraint. This defines a
generalization of the supersymmetric Casimir energy, whose Legendre transform
at large $N$ exactly reproduces the Bekenstein-Hawking entropy of the black
hole.
