Black Hole Entropy and the Dimensional Continuation of the Gauss-Bonnet Theorem

The Euclidean black hole has topology ${\mathrm{\Re }}^2\times {\mathcal{S}}^{d-2}$. It is shown that -in Einstein’s theory- the deficit angle of a cusp at any point in ${\mathrm{\Re }}^2$ and the area of the ${\mathcal{S}}^{d-2}$ are canonical conjugates. The black hole entropy emerges as the Euler class of a small disk centered at the horizon multiplied by the area of the ${\mathcal{S}}^{d-2}$ there.These results are obtained through dimensional continuation of the Gauss-Bonnet theorem. The extension to the most general action yielding second order field equations for the metric in any spacetime dimension is given.