  Versions of parameterized pseudo-Newtonian gravity theories specially
designed for cosmology have been introduced in recent cosmology literature. The
modifications demand a zero-pressure fluid in the context of versions of
modified Poisson-like equation with two different gravitational potentials. We
consider such modifications in the context of relativistic gravity theories
where the action is a general algebraic function of the scalar curvature, the
scalar field, and the kinetic term of the field. In general it is not possible
to isolate the zero-pressure fluid component simultaneously demanding a
modification in the Poisson-like equation. Only in the small-scale limit we can
realize some special forms of the attempted modifications. We address some
loopholes in the possibility of showing non-Einstein gravity nature based on
pseudo-Newtonian modifications in the cosmological context. We point out that
future observations of gravitational weak lensing together with velocity
perturbation can potentially test the validity of Einstein's gravity in
cosmology context.
