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We develop an effective superpotential formalism for the SU(2)$\times$U(1)invariant sector of $\mathcal{N}=2$ gauged supergravity in five dimensions witha U(1)$^3$ Fayet-Iliopoulos gauging, and determine the exact superpotentialthat describes all 1/4 BPS solutions in this sector. This includes theGutowski-Reall black holes, but also a much broader class of solutions with asquashed $S^3$, magnetic flux and vector multiplet sources, as well as complexEuclidean BPS saddles. Some of these solutions are known only numerically, butthe exact superpotential allows us to analytically evaluate the on-shellaction, holographic one-point functions and conserved charges of all BPSsolutions and to study their thermodynamics. In particular, by examining thesupersymmetry Ward identities we show that solutions with supersymmetric vectormultiplet sources break supersymmetry spontaneously. We also demonstrate thefirst law for black holes in the SU(2)$\times$U(1) invariant sector and showthat the conserved charges of supersymmetric solutions satisfy the generalizedBPS relation derived in arXiv:1703.04299, which includes the supersymmetricCasimir energy as a consequence of the anomalous supersymmetry transformationof the $\mathcal{N}=1$ supercurrent at the boundary. Finally, we show that theeffective superpotential provides a unifying entropy extremization principle,reproducing Sen's entropy function for near extremal black holes and theHosseini-Hristov-Zaffaroni functional for complex Euclidean BPS saddles.

Black hole superpotential as a unifying entropy function and BPS thermodynamics