A note on the definition of gravitational energy for quadratic curvature gravity via topological regularization

Within the framework of four-dimensional quadratic curvature gravities in the appearance of a negative cosmological constant, a definition for the gravitational energy of solutions with anti-de Sitter (AdS) asymptotics was put forward by Giribet et al. Phys. Rev. D 98 044046 (2018). This was achieved by adding proper topological invariant terms to the gravity action to render the variation problem well-posed. We prove that the definition via the procedure of topological regularization can be covered by our previous work Int. J. Mod. Phys. A 35 2050102 (2020) in four dimensions. Motivated by this, we further generalize the results to generic diffeomorphism invariant theories of gravity in arbitrary even dimensions.