  As an alternative to the Dirichlet counterterms prescription, I introduce the
concept of Kounterterms as the boundary terms with explicit dependence on the
extrinsic curvature K_{ij} that regularize the AdS gravity action. Instead of a
Dirichlet boundary condition on the metric, a suitable choice of the boundary
conditions --compatible with any asymptotically AdS (AAdS) spacetime-- ensures
a finite action principle for all odd dimensions. Background-independent
conserved quantities are obtained as Noether charges associated to asymptotic
symmetries and their general expression appears naturally split in two parts.
  The first one gives the correct mass and angular momentum for AAdS black
holes and vanishes identically for globally AdS spacetimes. Thus, the second
part is a covariant formula for the vacuum energy in AAdS spacetimes and
reproduces the results obtained by the Dirichlet counterterms method in a
number of cases. It is also shown that this Kounterterms series regularizes the
Euclidean action and recovers the correct black hole thermodynamics in odd
dimensions.
