  A finite action principle for Einstein-Gauss-Bonnet AdS gravity is presented.
The boundary term, which is different for even and odd dimensions, is a
functional of the boundary metric, intrinsic curvature and extrinsic curvature.
For even dimensions, the boundary term corresponds to the maximal Chern form of
the spacetime, and the asymptotic AdS condition for the curvature suffices for
the well-posedness of this action. For odd dimensions, the action is stationary
under a boundary condition on the variation of the extrinsic curvature. The
background-independent Noether charges associated to asymptotic symmetries are
found and the Euclidean continuation of the action correctly describes the
black hole thermodynamics in the canonical ensemble. In particular, this
procedure leads to a covariant formula for the vacuum energy in odd-dimensional
asymptotically AdS spacetimes.
