  A well-defined variational principle for gravitational actions typically
requires to cancel boundary terms produced by the variation of the bulk action
with a suitable set of boundary counterterms. This can be achieved by carefully
balancing the coefficients multiplying the bulk operators with those
multiplying the boundary ones. A typical example of this construction is the
Gibbons-Hawking-York boundary action that needs to be added to the
Einstein-Hilbert one in order to have a well-defined metric variation for
General Relativity with Dirichlet boundary conditions. Quantum fluctuations of
matter fields lead to the renormalization of said coefficients which may or may
not preserve this balance. Indeed, already at the level of General Relativity,
the resilience of the matching between bulk and boundary constants is far from
obvious and it is anyway incomplete given that matter generically induces
quadratic curvature operators. We investigate here the resilience of the
matching of higher-order couplings upon renormalization by a non-minimally
coupled scalar field and show that a problem is present. Even though we do not
completely solve the latter, we show that it can be greatly ameliorated by a
wise splitting between dynamical and topological contributions. Doing so, we
find that the bulk-boundary matching is preserved up to a universal term, whose
nature and possible cancellation we shall discuss in the end.
