  In this paper, the connection between the Lorentz-covariant counterterms that
regularize the four-dimensional AdS gravity action and topological invariants
is explored. It is shown that demanding the spacetime to have a negative
constant curvature in the asymptotic region permits the explicit construction
of such series of boundary terms. The orthonormal frame is adapted to
appropriately describe the boundary geometry and, as a result, the boundary
term can be expressed as a functional of the boundary metric, extrinsic
curvature and intrinsic curvature. This choice also allows to write down the
background-independent Noether charges associated to asymptotic symmetries in
standard tensorial formalism. The absence of the Gibbons-Hawking term is a
consequence of an action principle based on a boundary condition different than
Dirichlet on the metric. This argument makes plausible the idea of regarding
this approach as an alternative regularization scheme for AdS gravity in all
even dimensions, different than the standard counterterms prescription. As an
illustration of the finiteness of the charges and the Euclidean action in this
framework, the conserved quantities and black hole entropy for four-dimensional
Kerr-AdS are computed.
