The evolutionary behavior over cosmological time scales of hot intracluster gas in rich clusters of galaxies is examined with a spherically symmetric hydrodynamical code, for the important case of a time-dependent cluster gravitational potential. Both infall of primordial matter and ejection of metal-enriched galactic material are considered. Cluster X-ray properties as a function of redshift are predicted, and the results are compared to observations of low-redshift clusters. The effects of evolution are strong; for example, cluster X-ray luminosities increase by about a factor of 10 from the epoch with redshift z = 1 until the present. It is shown that the effects of evolution dominate in any cosmological test with X-ray clusters.