  The renormalisation group improved Standard Model effective potential in an
arbitrary curved spacetime is computed to one loop order in perturbation
theory. The loop corrections are computed in the ultraviolet limit, which makes
them independent of the choice of the vacuum state and allows the derivation of
the complete set of $\beta$-functions. The potential depends on the spacetime
curvature through the direct non-minimal Higgs-curvature coupling, curvature
contributions to the loop diagrams, and through the curvature dependence of the
renormalisation scale. Together, these lead to significant curvature
dependence, which needs to be taken into account in cosmological applications,
which is demonstrated with the example of vacuum stability in de Sitter space.
