  It is well-known that the presence of a spacetime boundary requires the
conventional Einstein-Hilbert (EH) action to be supplemented by the
Gibbons-Hawking (GH) boundary term in order to retain the standard variational
procedure. When the EH action is amended by the diffeomorphism-invariant
graviton mass and potential terms, it naively appears that no further boundary
terms are needed since all the new fields of massive gravity enter the action
with the first derivative. However, we show that such a formulation would be
inconsistent, even when the bulk action is ghost-free. The theory is
well-defined only after introducing novel boundary counterterms, which dominate
over the GH term in the massless limit and cancel the problematic boundary
terms induced by the bulk action. The number of boundary counterterms equals
the number of total derivatives one could construct in the bulk using positive
powers of two derivatives of the longitudinal mode of the massive graviton.
