Does the expanding universe really have an effect on the Higgs field? Say, is it like the matter getting diluted with the expanding universe? Or does the expansion does not have any effect on the Higgs field, just like the Dark Energy maintaining its energy uniform even if the universe is expanding?
At the moment there does not exist a quantized unified theory of gravitation and the standard model. The only candidate to date is string theories which are at a research level, and in which I am not able to form an answer.
There exist effective quantizations of gravity and effective models including the standard model in the studies of cosmology and the Big Bang model. In these models the continuing expansion of the universe is not affecting the bound states of the universe, atoms, molecules, gravitational masses, galaxies , clusters. Even the gravitational field is much stronger then the effective "force" of expansion. ( the raisin bread analogue for the expansion and the stability of bound states).
The Higgs field, once the energies became low enough to break the symmetry which gives zero masses at very high energies during the original expansion, belongs to the set of fields which defines the bound states of atoms and molecules, together with the strong weak and electromagnetic fields. Thus the Higgs field will also not be affected by the continuing expansion of the universe.
This is a hand waving explanation within the known theories, which needs a theorist's input , imo. See one such by Lubos Motl .