  Among many alternative gravitational theories to General Relativity (GR),
$f(R,T)$ gravity (where $R$ is the Ricci scalar and $T$ the trace of the
energy-momentum tensor) has been widely studied recently. By adding a matter
contribution to the gravitational Lagrangian, $f(R,T)$ theories have become an
interesting extension to GR displaying a broad phenomenology in astrophysics
and cosmology. In this paper, we discuss however the difficulties appearing in
explaining a viable and realistic cosmology within the $f(R,T)$ class of
theories. Our results challenge the viability of $f(R,T)$ as an alternative
modification of gravity.
