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 λ=1/2. The unimodular gravity proposal is a very interesting approach in order 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 Tμν;μ=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.