  Extensions of the gravity theory in order to obtain traceless field equations
have been widely considered in the literature. The leading example of such
class of theories is the unimodular gravity, but there are other possibilities
like the mimetic gravity and the Rastall gravity with a coupling parameter
$\lambda = 1/2$. The unimodular gravity proposal is a very interesting approach
in other to address the cosmological constant problem. When coupled to matter
such theories may imply that the energy-momentum tensor is not divergence free
anymore. In this paper, a unimodular type theory will be developed by evading
the conservation $T^{\mu\nu}_{\, ; \mu}=0$. The cosmological consequences of
the later, both at background as well as for scalar and tensor perturbations,
are explored. Possible further extensions of this approach are discussed as
well as its connection with the traditional unimodular gravity.
