  We investigate the effects of noncommutativity between the position-position,
position-momentum and momentum-momentum of a phase space corresponding to a
modified cosmological model. We show that the existence of such
noncommutativity results in a Moyal Poisson algebra between the phase space
variables in which the product law between the functions is of the kind of an
$\alpha$-deformed product. We then transform the variables in such a way that
the Poisson brackets between the dynamical variables take the form of a usual
Poisson bracket but this time with a noncommutative structure. For a power law
expression for the function of the Ricci scalar with which the action of the
gravity model is modified, the exact solutions in the commutative and
noncommutative cases are presented and compared. In terms of these solutions we
address the issue of the late time acceleration in cosmic evolution.
