  We investigate the specific heat at constant volume $C_V$ in the BCS
(Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover regime
of an ultracold Fermi gas above the superfluid phase transition temperature
$T_{\rm c}$. Within the framework of the strong-coupling theory developed by
Nozi\`eres and Schmitt-Rink, we show that this thermodynamic quantity is
sensitive to the stability of preformed Cooper pairs. That is, while
$C_V(T\gesim T_{\rm c})$ in the unitary regime is remarkably enhanced by {\it
metastable} preformed Cooper pairs or pairing fluctuations, it is well
described by that of an ideal Bose gas of long-lived {\it stable} molecules in
the strong-coupling BEC regime. Using these results, we identify the region
where the system may be viewed as an almost ideal Bose gas of stable pairs, as
well as the pseudogap regime where the system is dominated by metastable
preformed Cooper pairs, in the phase diagram of an ultracold Fermi gas with
respect to the strength of a pairing interaction and the temperature. We also
show that the calculated specific heat agrees with the recent experiment on a
$^6$Li unitary Fermi gas. Since the formation of preformed Cooper pairs is a
crucial key in the BCS-BEC crossover phenomenon, our results would be helpful
in considering how fluctuating preformed Cooper pairs appear in a Fermi gas, to
eventually become stable, as one passes through the BCS-BEC crossover region.
