  We use numerical methods to exhaustively study a novel family of hairy black
hole solutions in AdS$_5$. These solutions can be uplifted to solutions of type
IIB supergravity with AdS$_5\times S^5$ asymptotics and are thus expected to
play an important role in our understanding of AdS/CFT. We find an intricate
phase diagram, with the aforementioned family of hairy black hole solutions
branching from the Reissner-Nordstr\"om black hole at the onset of the
superradiance instability. We analyse black holes with spherical and planar
horizon topology and explain how they connect in the phase diagram. Finally, we
detail their global and local thermodynamic stability across several ensembles.

High Energy Physics - Theory;General Relativity and Quantum Cosmology