  The polytropic equation of state (EOS) of primordial gas clouds with modest
enrichment is computed, motivated by the recent observations of very
Fe-deficient stars, [Fe/H]~10^{-3.5}-10^{-5}, such as HE 0107-5240 and CS
29498-043. These stars are overabundant, relative to Fe, in C and O. We assume
that the observed abundances of species like C, O, Si and Fe are representative
of the gas from which the currently observed metal-deficient stars formed.
Under this assumption, we find that this primordial metal abundance pattern has
profound consequences for the thermal balance and chemical composition of the
gas, and hence for the EOS of the parental cloud. The polytopic EOS is soft for
low, [O/H]<10^{-3}, oxygen abundances, but stiffens to a polytropic index gamma
large than unity for [O/H]>10^{-2} due to the large opacity in the CO and H2O
cooling lines. It is further found that a regulating role is played by the
presence and temperature of the dust, even when the overall carbon abundance is
only [C/H]~10^{-2}. When the dust is warmer than the gas, a region with
gamma~1.2 results around a density of ~10^4 cm^{-3}. When the dust is colder
than the gas, a region with gamma~0.8 is found for a density of ~10^6 cm^{-3}.
Implications for the primordial initial mass function (IMF) as well as the IMF
in starburst galaxies, where the metallicity is super-solar, are explored and
related to processes that influence the temperature of the ambient dust.
