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Universal Spinning Black Holes and Theories of Class $\mathcal R$

We study a supersymmetric, rotating, electrically charged black hole in AdS$_{4}$ which is a solution of four-dimensional minimal gauged supergravity. Using holography we show that the free energy on $S^3$ and the superconformal index of the dual three-dimensional $\mathcal N=2$ SCFT, in the planar limit, are related in a simple universal way. This result applies to large classes of SCFTs constructed from branes in string and M-theory which we discuss in some detail. For theories of class $\mathcal R$, which arise from $N$ M5-branes wrapped on hyperbolic three-manifolds, we show that the superconformal index agrees with the black hole entropy in the large $N$ limit.