Recent observations of the entropy of the intracluster medium (ICM) at high redshift are not consistent with a redshift-independent entropy floor. To interpret these observations, we study models in which the entropy of ICM is enhanced at the epoch of cluster formation (zf), which then passively evolves as a result of radiative cooling until the epoch of observation (zo). We confirm that a z-independent entropy floor is incompatible with the observations. We find that models in which energy deposition into the ICM increases with redshift [scaling approximately as ∼M0.6cl(1 +zf)2.2] are consistent with observations of low- and high-redshift clusters. Possible sources of such non-gravitational heating are briefly discussed.