  There has been recent interest in the cosmological consequences of
energy-momentum-powered gravity models, in which the matter side of Einstein's
equations is modified by the addition of a term proportional to some power,
$n$, of the energy-momentum tensor, in addition to the canonical linear term.
In this work we treat these models as phenomenological extensions of the
standard $\Lambda$CDM, containing both matter and a cosmological constant. We
also quantitatively constrain the additional model parameters using low
redshift background cosmology data that are specifically from Type Ia
supernovas and Hubble parameter measurements. We start by studying specific
cases of these models with fixed values of $n,$ which lead to an analytic
expression for the Friedmann equation; we discuss both their current
constraints and how the models may be further constrained by future
observations of Type Ia supernovas for WFIRST complemented by measurements of
the redshift drift by the ELT. We then consider and constrain a more extended
parameter space, allowing $n$ to be a free parameter and considering scenarios
with and without a cosmological constant. These models do not solve the
cosmological constant problem per se. Nonetheless these models can
phenomenologically lead to a recent accelerating universe without a
cosmological constant at the cost of having a preferred matter density of
around $\Omega_M\sim0.4$ instead of the usual $\Omega_M\sim0.3$. Finally we
also briefly constrain scenarios without a cosmological constant, where the
single component has a constant equation of state which needs not be that of
matter; we provide an illustrative comparison of this model with a more
standard dynamical dark energy model with a constant equation of state.
