Universal renormalization procedure for higher curvature gravities in D$\leq$5

We implement a universal method for renormalizing AdS gravity actions applicable to arbitrary higher curvature theories in up to five dimensions. The renormalization procedure considers the extrinsic counterterm for Einstein-AdS gravity given by the Kounterterms scheme, but with a theory-dependent coupling constant that is fixed by the requirement of renormalization for the vacuum solution. This method is shown to work for a generic higher curvature gravity with arbitrary couplings except for a zero measure subset, which includes well-known examples where the asymptotic behavior is modified and the AdS vacua are degenerate, such as Chern-Simons gravity in 5D, Conformal Gravity in 4D and New Massive Gravity in 3D. In order to show the universality of the scheme, we perform a decomposition of the equations of motion into their normal and tangential components with respect to the Poincare coordinate and study the Fefferman-Graham expansion of the metric. We verify the cancellation of divergences of the on-shell action and the well-posedness of the variational principle.