Entropy and topology for gravitational instantons

In this work a relation between topology and thermodynamical features of gravitational instantons is shown. The expression for the Euler characteristic, through the Gauss-Bonnet integral, and the one for the entropy of gravitational instantons are proposed in a form that makes the relation between them self-evident. A new formulation of the Bekenstein-Hawking formula, where the entropy and the Euler characteristic are related by $S=\chi A/8,$ is obtained. This formula provides the correct results for a wide class of gravitational instantons described by both spherically and axially symmetric metrics.