  The Euclidean black hole has topology $\Re^2 \times {\cal S}^{d-2}$. It is
shown that -in Einstein's theory- the deficit angle of a cusp at any point in
$\Re^2$ and the area of the ${\cal S}^{d-2}$ 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 ${\cal S}^{d-2}$ 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.
