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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, $T=ρ-3p$, 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 $Λ$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 $Λ$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 $Λ$CDM limit) are ruled out. This provides some insight on the level of robustness of the $Λ$CDM model and on the parameter space still available for phenomenological alternatives and extensions.