Odd-dimensional localization in supergravity

We establish a localization principle for odd-dimensional supergravity theories containing Chern--Simons interactions. Our construction relies on combining an equivariant extension of the gauge-invariant part of the action with an equivariant completion of the Chern--Simons anomaly form $Φ^{(\text{anom})}$ to form a relative equivariant cohomology class. This leads to a universal fixed point formula for the on-shell action of supersymmetric solutions that is expressed in terms of $Φ^{(\text{anom})}$ and an equivariant Euler class. We illustrate the formalism in $D=11$ and $D=5$ supergravity, with applications including the supersymmetric Casimir energy and superconformal index of M5-branes, as well as black ring and black lens solutions.