  We study the Newtonian cosmology taking into account the leading classical
and quantum corrections of order $\mathcal{O}(G^{2})$ in the Newtonian
potential. We first derive the modified Friedmann equations starting from the
non-relativistic conservation of kinetic energy and potential energy for an
infinitesimal mass. We then consider the leading classical correction term and
the quantum correction term in the Newtonian potential for deriving the
Friedmann equation, however, the quantum correction term is too small and hence
does not contribute in the physical results. We then investigate the difference
in scale factor with the usual scale factor for various matter like radiation,
dust and cosmological constant by considering the corrections in the Newtonian
potential. We observe that the evolution of the universe is similar for
radiation and dust cases at late times. The cosmological constant case shows a
steep increase in the scale factor compared to the other cases. We also note
that the universe may have a bounce in the case of radiation depending on the
sign of the coefficient of the leading classical correction.
