  We theoretically investigate the specific heat $C_V$ at constant volume in
the normal state of a $p$-wave interacting Fermi gas. Including fluctuations in
the $p$-wave Cooper channel within the framework of the strong-coupling theory
developed by Nozi\`eres and Schmitt-Rink, we clarify how $C_V$ as a function of
temperature varies, as one moves from the weak-coupling regime to the
strong-coupling limit. In the weak-coupling regime, $C_V$ is shown to be
enhanced by $p$-wave pairing fluctuations, near the superfluid phase transition
temperature $T_{\rm c}$. Similar enhancement of $C_V(T\simeq T_{\rm c})$ is
also obtained in the strong-coupling regime, which, however, reflects that
system is close an ideal Bose gas of $p$-wave two-body bound molecules. Using
these results, we classify the normal state into (1) the normal Fermi gas
regime, (2) the $p$-wave molecular Bose gas regime, and (3) the region between
the two, where $p$-wave pairing fluctuations are dominant. Since the current
experiments can only access the normal phase of a $p$-wave interacting Fermi
gas, our results would be useful for experiments to understand strong-coupling
properties of this Fermi system above $T_{\rm c}$.
