  We develop a cosmological model based on action-dependent Lagrangian
theories. The main feature here is the nonconservation of the energy momentum
tensor due to the nontrivial geometrical construction of the theory. We provide
the basic set of equations necessary to study both the cosmological background
expansion as well as the linear matter perturbation growth. We show that the
simplest realization of the Universe as described by only one component is not
viable as expected from the existing correspondence between this model and the
case of viscous cosmological fluids. However, modeling the energy content of
the Universe as composed by two pressureless fluids, i.e., one a typical cold
dark matter fluid and the other a pressureless dark energy fluid which is
responsible for driving the late-time acceleration expansion, is qualitatively
compatible with observational data.
