  The Hamiltonian formalism of the generalized unimodular gravity theory, which
was recently suggested as a model of dark energy, is shown to be a complicated
example of constrained dynamical system. The set of its canonical constraints
has a bifurcation -- splitting of the theory into two branches differing by the
number and type of these constraints, one of the branches effectively
describing a gravitating perfect fluid with the time-dependent equation of
state, which can potentially play the role of dark energy in cosmology. The
first class constraints in this branch generate local gauge symmetries of the
Lagrangian action -- two spatial diffeomorphisms -- and rule out the temporal
diffeomorphism which does not have a realization in the form of the canonical
transformation on phase space of the theory and turns out to be either nonlocal
in time or violating boundary conditions at spatial infinity. As a consequence,
the Hamiltonian reduction of the model enlarges its physical sector from two
general relativistic modes to three degrees of freedom including the scalar
graviton. This scalar mode is free from ghost and gradient instabilities on the
Friedmann background in a wide class of models subject to a certain restriction
on time-dependent parameter $w$ of the dark fluid equation of state,
$p=w\varepsilon$. For a special family of models this scalar mode can be ruled
out even below the phantom divide line $w=-1$, but this line cannot be crossed
in the course of the cosmological expansion. This is likely to disable the
generalized unimodular gravity as a model of the phenomenologically consistent
dark energy scenario, but opens the prospects in inflation theory with a scalar
graviton playing the role of inflaton.
