  In this work we review the issue of imposing the conservation of the
energy-momentum tensor as a necessary condition to recover the equivalence
between the unimodular gravity and General Relativity (GR) equipped with a
cosmological constant. This procedure is usually interpreted as an {\it ad hoc}
imposition on the unimodular theory's structure. Whereas the consequences of
avoiding the conservation of the total energy-momentum tensor has been already
introduced in the literature, it has been not widely explored so far. We study
an expanding universe sourced by a single effective perfect fluid such that the
null divergence of its energy-momentum tensor is not imposed. As we shall show,
in this scheme, the unimodular theory has its own conservation equation
obtained from the Bianchi identities. We explore the evolution of the
homogeneous and isotropic expanding background and show that a viable
cosmological scenario exists. Also, we consider scalar perturbations with
particular attention given to the gauge issue. We show that contrary to the
traditional unimodular theory where the synchronous and longitudinal
(newtonian) gauge for cosmological perturbations are not permitted, if the
conservation of the energy-momentum is relaxed the scalar perturbations in the
synchronous condition survive and present a growing mode behavior. We study
therefore a new cosmological scenario in which the dynamics of the universe
transits from the radiative phase directly to a accelerated one but allowing
thus for structure formation.
