  A new class of modified gravity theories with a healthy higher order
derivative terms of a function of the matter Lagrangian $f(L_m)$ is considered.
Generally the energy momentum tensor is not conserved, leading to the fifth
force similar to $f(R,T)$ theories. We will however show that in the FRW
background there exists two possibilities corresponding to conservative and
non-conservative cases. Cosmological implications of both cases with different
functions of the matter Lagrangian $f$ will be investigated in details and we
will show that current observational data can be satisfied in this model. The
non-conservative case however, predicts less matter sources at early times and
more deceleration. Evolution of the matter density perturbation in the
longitudinal gauge is also considered for dust matter sources. We will show
that the late time behavior of the matter density perturbation is in agreement
with current observational data at sub-horizon limit.
