  We find numerically that the sample to sample fluctuation of the entropy,
$\Delta S$, is a tool more sensitive in distinguishing how from high
temperature behaviors, than the corresponding fluctuation in the free energy.
In 1+1 dimensions we find a single phase for all temperatures since $(\Delta
S)^{2}$ is always extensive. In 2+1 dimensions we find a behavior may look at
first sight as a transition from a low temperature phase where $(\Delta S)^{2}$
is extensive to a high temperature phase where it is subextensive. This is
observed in spite of the relatively large system we use. The observed behavior
is explained not as a phase transition but as a strong crossover behavior. We
use an analytical agreement to obtain $(\Delta S)^{2}$ for high temperature and
find that while it is always extensive it is also extremly small and the
leading extensive part decays very fast to zero as temperature is increased.
