In a class of models designed to solve the cosmological constant problem by coupling scalar or tensor classical fields to the space-time curvature, the universal scale factor grows as a power law in the age, $a propto t^alpha$, regardless of the matter content or cosmological epoch. We investigate constraints on such power-law cosmologies from the present age of the Universe, the magnitude-redshift relation, and from primordial nucleosynthesis. Constraints from the current age of the Universe and from the high-redshift supernovae data require large $alpha$ ($approx 1$), while consistency with the inferred primordial abundances of deuterium and helium-4 forces $alpha$ to lie in a very narrow range around a lower value ($approx 0.55$). Inconsistency between these independent cosmological constraints suggests that such power-law cosmologies are not viable.
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