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We study the Hamiltonian structure of unimodular-like theories, where thecosmological constant (or other supposed constants of nature) are demoted fromfixed parameters to classical constants of motion. No new local degrees offreedom are present as a result of a $U(1)$ gauge invariance of the theory.Hamiltonian analysis of the action reveals that the only possible gauge fixingthat can be enforced is setting the spatial components of the four-volume timevector ${\cal T}^{i}\approx0$. As a consequence of this, the gauge-fixedunimodular path integral is equivalent to the minisuperspace unimodular pathintegral. However, should we break the $U(1)$ gauge invariance, two thingshappen: a massless propagating degree of freedom appears, and the(gauge-invariant) zero-mode receives modified dynamics. The implications areinvestigated, with the phenomenology depending crucially on the target``constant''.

Unimodular Proca Theory: Breaking the U(1) gauge symmetry of unimodular gravity via a mass term