  In this paper, we fully investigate the cosmological effects of the moduli
dependent one-loop corrections to the gravitational couplings of the string
effective action to explain the cosmic acceleration problem in early (and/or
late) universe. These corrections comprise a Gauss-Bonnet (GB) invariant
multiplied by universal non-trivial functions of the common modulus $\sigma$
and the dilaton $\phi$. The model exhibits several features of cosmological
interest, including the transition between deceleration and acceleration
phases. By considering some phenomenologically motivated ansatzs for one of the
scalars and/or the scale factor (of the universe), we also construct a number
of interesting inflationary potentials. In all examples under consideration, we
find that the model leads only to a standard inflation ($w \geq -1$) when the
numerical coefficient $\delta$ associated with modulus-GB coupling is positive,
while the model can lead also to a non-standard inflation ($w<-1$), if $\delta$
is negative. In the absence of (or trivial) coupling between the GB term and
the scalars, there is no crossing between the $w< -1$ and $w> -1$ phases, while
this is possible with non-trivial GB couplings, even for constant dilaton phase
of the standard picture. Within our model, after a sufficient amount of e-folds
of expansion, the rolling of both fields $\phi$ and $\sigma$ can be small. In
turn, any possible violation of equivalence principle or deviations from the
standard general relativity may be small enough to easily satisfy all
astrophysical and cosmological constraints.
