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Recently, a practical approach to holographic renormalization has beendeveloped based on the Hamilton-Jacobi formulation. Using a simpleEinstein-scalar theory, we clarify that this approach does not conflict withthe Hamiltonian constraint as it seems. Then we apply it to the holographicrenormalization of massive gravity. We assume that the shift vector is fallingoff fast enough asymptotically. We derive the counterterms up to the boundarydimension d=4. Interestingly, we find that the conformal anomaly can even occurin odd dimensions, which is different from the Einstein gravity. We check thatthe counterterms cancel the divergent part of the on-shell action at thebackground level. At the perturbation level, they are also applicable inseveral time-dependent cases.

Hamilton-Jacobi Approach to Holographic Renormalization of Massive Gravity