  In a superstring framework, the free energy density, F, can be determined
unambiguously at the full string level once supersymmetry is spontaneously
broken via geometrical fluxes. We show explicitly that only the moduli
associated to the supersymmetry breaking may give relevant contributions. All
other spectator moduli \mu_I give exponentially suppressed contributions for
relatively small (as compared to the string scale) temperature, T, and
supersymmetry breaking scale, M. More concisely, for \mu_I > T and M, F takes
the form
  F(T,M; \mu_I)=F(T,M)+O[exp(- {\mu_I\over T}), exp(- {\mu_I\over M})]
  We study the cosmological regime where T and M are below the Hagedorn
temperature scale T_H. In this regime, F remains finite for any values of the
spectator moduli \mu_I. We investigate extensively the case of one spectator
modulus \mu_d corresponding to R_d, the radius-modulus field of an internal
compactified dimension. We show that its thermal effective potential admits
five phases, each of which can be described by a distinct but different
effective field theory. For late cosmological times, the Universe is attracted
to a "Radiation-like evolution" with M(t) ~ T(t)~ 1/a(t)~ t^{-2/d}. The
spectator modulus \mu(t) is stabilized either to the stringy enhanced symmetry
point where R_d=1, or fixed at an arbitrary constant \mu_0>T,M. For arbitrary
boundary conditions at some initial time, t_E, \mu(t) may pass through more
than one effective field theory phase before its final attraction.
