  We calculate the one-loop effective potential of a scalar field in a
Robertson-Walker background with scalar metric perturbations. A complete set of
orthonormal solutions of the perturbed equations is obtained by using the
adiabatic approximation for comoving observers. After analyzing the problem of
renormalization in inhomogeneous backgrounds, we get the explicit contribution
of metric perturbations to the effective potential. We apply these results to
the Standard Model Higgs field and evaluate the effects of metric perturbations
on the Higgs mass and on its vacuum expectation value. Space-time variations
are found, which are proportional to the gravitational slip parameter, with a
typical amplitude of the order of $\Delta\phi/\phi\simeq 10^{-11}$ on
cosmological scales. We also discuss possible astrophysical signatures in the
Solar System and in the Milky Way that could open new possibilities to explore
the symmetry breaking sector of the electroweak interactions.
