Equivariant localization in supergravity in odd dimensions

We discuss a localization formula for certain integrals on odd-dimensional manifolds with boundaries, equipped with a Killing vector, and employ this to localize the regularised on-shell action of a large class of supersymmetric solutions of five dimensional minimal gauged supergravity. Specifically, we consider asymptotically AdS_5 solutions in the time-like class, in which the transverse Kähler foliation is assumed to be toric. We find that the background subtraction regularization method leads to an intriguing formula for the on-shell action, in terms of an analytic continuation of the Martelli-Sparks-Yau Sasakian volume. In particular, we show that the regularised on-shell action is a function of the toric data of an effective compact five-dimensional manifold, as well as of the supersymmetric Killing vector, outside the corresponding dual cone. As our main example we provide a derivation of the well-known entropy function of supersymmetric and rotating black holes in AdS_5, using only topological data.