The black hole at the end of the cone: localizing the anomaly polynomial on toric geometries

We consider five-dimensional supergravity coupled to vector multiplets, gauged or ungauged, and propose an efficient method to evaluate the on-shell action of supersymmetric black saddle solutions with toric $\mathrm{U}(1{)}^3$ symmetry and general topology, explicitly known or just assumed to exist, including higher-derivative corrections. This is equivalent to equivariant integration of the anomaly polynomial six-form over simplicial cones obtained by decomposing the toric diagram of the solution. The on-shell action is then expressed as a sum over contributions localized at the tip of each cone. We obtain a simple derivation of recently calculated expressions as well as new predictions, both for the on-shell action of the black saddles and the Wald entropy of the related extremal solutions. These may be asymptotically AdS${}_5$ or asymptotically flat. As examples, we discuss black holes, black rings and black lenses, including a black hole in the background of a topological soliton.