  We study the Hamiltonian structure of unimodular-like theories, where the
cosmological constant (or other supposed constants of nature) are demoted from
fixed parameters to classical constants of motion. No new local degrees of
freedom 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 fixing
that can be enforced is setting the spatial components of the four-volume time
vector ${\cal T}^{i}\approx0$. As a consequence of this, the gauge-fixed
unimodular path integral is equivalent to the minisuperspace unimodular path
integral. However, should we break the $U(1)$ gauge invariance, two things
happen: a massless propagating degree of freedom appears, and the
(gauge-invariant) zero-mode receives modified dynamics. The implications are
investigated, with the phenomenology depending crucially on the target
``constant''.
