For many even-dimensional supergravities, the on-shell action of a generic supersymmetric solution has been computed with equivariant localization techniques, and the result only depends on topological information. For odd-dimensional supergravities a similar computation is more challenging, since Chern-Simons terms make the Lagrangian gauge-dependent and the supersymmetric Killing vector typically has no fixed points. In this talk, I will explain how these issues can be resolved, and I will present a formula for the localized on-shell action of a large class of solutions of 5D supergravities, including black holes, black rings, black lenses, various topological solitons and more.
