| Summary: | <p>We introduce a general class of toric gravitational instantons in <em>D</em> = 4, <em>N</em> = 2 gauged supergravity, namely Euclidean supersymmetric solutions with <em>U</em>(1)<sup>2</sup> isometry. Such solutions are specified by a “supergravity labelled polytope”, where the labels encode the 4-manifold topology, the choice of magnetic fluxes, and certain signs associated to the Killing spinor. Equivariant localization allows us to write down the gravitational free energy for such a solution, assuming it exists, and study its properties. These results open the way for a systematic study of holography in this setting, where the dual large N field theories are defined on the boundary 3-manifolds, which are (squashed) lens spaces <em>L</em>(<em>p</em>, <em>q</em>) or generalizations with orbifold singularities.</p>
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