  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.
