Domain of validity of the evolution of perturbations in Newtonian cosmology with pressure

It has been argued that by removing a pressure gradient term in the continuity equation, it is possible to obtain, with a semiclassical formulation, the same expression for the density contrast growing mode, as obtained in a full relativistic treatment. In this context, we reinvestigate the evolution of perturbations in an expanding Newtonian universe with pressure, but we consider a general scenario in which the equation of state parameter is time dependent and the perturbations are not necessarily adiabatic. We verify that, in this case, the suggested modification of the continuity equation does not provide equivalence between the relativistic and Newtonian descriptions.