| In a superstring framework, the free energy density F can be determined unambiguously 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 temperature T and supersymmetry 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/T), exp(-\mu_I/M) ]. For T and M below the Hagedorn temperature scale T_H, F remains finite for any values of the spectator moduli \mu_I. We investigate extensively the case of one spectator modulus \mu ~ 1/(R_s+1/R_s), with R_s the radius-modulus field of an internal compactified dimension. We show that its thermal effective potential V(T,M; \mu) = -F(T,M; \mu) admits five phases, each of which can be described by a distinct 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_s=1, or fixed at an arbitrary constant \mu_0>T,M. If \mu(t_E) |
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