Holographic Counterterms from Restricted Weyl Symmetry in Even-Dimensional Einstein-AdS Gravity

We develop a symmetry-based approach to renormalization for even-dimensional Einstein-AdS gravity based on infinitesimal bulk Weyl transformations that preserve its field equations asymptotically. We determine a bulk-covariant completion of the Einstein-Hilbert density by requiring its variation under restricted Weyl transformations with parameter $Σ(X)$ to vanish modulo terms whose integrals remain finite as the radial cutoff is removed. This requirement induces two types of compensating terms with fixed couplings: the Euler density and a hierarchy of covariant total derivatives. The resulting recursive relations admit a factorized solution in terms of shifted Laplacians, yielding a closed-form completion in arbitrary even bulk dimensions. On Einstein backgrounds, these expressions agree, after matching normalizations, with those obtained through the ambient construction of conformal invariants. The associated renormalized-volume identity establishes on-shell finiteness for Euclidean AlAdS Einstein geometries in arbitrary even dimensions. The prescription reproduces the known conformally renormalized actions in four, six and eight dimensions and provides an explicit ten-dimensional completion involving powers of the Laplacian up to ³. Applications to eight- and ten-dimensional AdS black holes with non-conformally-flat Einstein horizons yield finite on-shell actions in agreement with Holographic Renormalization.