Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem

The Euclidean black hole has topology gerR^2scrS^d-2. It is shown that, in Einstein's theory the deficit angle of a cusp at any point in gerR^2 and the area of the scrS^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 scrS^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.