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.

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