  We propose a modified theory of gravitation constructed by the addition of
the term $f(T_{\mu\nu}T^{\mu\nu})$ to the Einstein-Hilbert action, and
elaborate a particular case
$f(T_{\mu\nu}T^{\mu\nu})=\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$, where $\alpha$
and $\eta$ are real constants, dubbed as energy-momentum powered gravity
(EMPG). We search for viable cosmologies arising from EMPG especially in the
context of the late-time accelerated expansion of the Universe. We investigate
the ranges of the EMPG parameters $(\alpha,\eta)$ on theoretical as well as
observational grounds leading to the late-time acceleration of the Universe
with pressureless matter only, while keeping the successes of standard general
relativity at early times. We find that $\eta=0$ corresponds to the
$\Lambda$CDM model, whereas $\eta\neq 0$ leads to a $w$CDM-type model. However,
the underlying physics of the EMPG model is entirely different in the sense
that the energy in the EMPG Universe is sourced by pressureless matter only.
Moreover, the energy of the pressureless matter is not conserved, namely, in
general it does not dilute as $\rho\propto a^{-3}$ with the expansion of the
Universe. Finally, we constrain the parameters of an EMPG-based cosmology with
a recent compilation of 28 Hubble parameter measurements, and find that this
model describes an evolution of the Universe similar to that in the
$\Lambda$CDM model. We briefly discuss that EMPG can be unified with
Starobinsky gravity to describe the complete history of the Universe including
the inflationary era.
