  We employ the techniques of the Functional Renormalization Group in string
theory, in order to derive an effective mini-superspace action for cosmological
backgrounds to all orders in the string scale $\alpha'$. To this end, T-duality
plays a crucial role, classifying all perturbative curvature corrections in
terms of a single function of the Hubble parameter. The resulting
renormalization group equations admit an exact, albeit non-analytic, solution
in any spacetime dimension $D$, which is however incompatible with Einstein
gravity at low energies. Within an $\epsilon$-expansion about $D=2$, we also
find an analytic solution which exhibits a non-Gaussian ultraviolet fixed point
with positive Newton coupling, as well as an acceptable low-energy limit. Yet,
within polynomial truncations of the full theory space, we find no evidence for
an analog of this solution in $D=4$. Finally, we comment on potential
cosmological implications of our findings.
