  In this note we explore a modified theory of gravitation that is not based on
the least action principle, but on a natural generalization of the original
Einstein's field equations. This approach leads to the non-covariant
conservation of the stress-energy tensor, a feature shared with other
Lagrangian theories of gravity such as the $f(R,T)$ case. We consider the
cosmological implications of a pair of particular models within this theory,
and we show that they have some interesting properties. In particular, for some
of the studied models we find that the density is bounded from above, and
cannot exceed a maximum value that depends on certain physical constants. In
the last part of the work we compare the theory to the $f(R,T)$ case and show
that they lead to different predictions for the motion of test particles.
