  The search for the physical mechanism underlying the observational evidence
for the acceleration of the recent universe is a compelling goal of modern
fundamental cosmology. Here we quantitatively study a class of homogeneous and
isotropic cosmological models in which the matter side of Einstein's equations
includes, in addition to the canonical term, a term proportional to the trace
of the energy-momentum tensor, $T=\rho-3p$, and constrain these models using
low redshift background cosmology data. One may think of these models as
extensions of general relativity with a nonlinear matter Lagrangian, and they
can be studied either as phenomenological extensions of the standard
$\Lambda$CDM model, containing both matter and a cosmological constant, or as
direct alternatives to it, where there is no cosmological constant but the
additional terms would have to be responsible for accelerating the universe.
Overall, our main finding is that parametric extensions of $\Lambda$CDM are
tightly constrained, with additional model parameters being constrained to
their canonical behaviours to within one standard deviation, while alternative
models in this class (which do not have a $\Lambda$CDM limit) are ruled out.
This provides some insight on the level of robustness of the $\Lambda$CDM model
and on the parameter space still available for phenomenological alternatives
and extensions.
