The heat capacity for a system of nucleons as a function of temperature is determined from the canonical partition function, which is projected from an approximation to the grand canonical partition function. This approximation is obtained from a functional integral representation of the partition function, including both thermodynamic fluctuations and first order quantum corrections. The number projection methods include a simple contour integration of ``Wick rotation'', and a Chebyschev polynomial expansion, both of which are formally exact. The above procedure is tested in a simple
i13/2 pairing model for which the exact canonical and grand canonical results can be obtained. It is found that both number projection methods fail to predict the position or the height of the peak in the heat capacity due to errors in the approximation to the grand partition function, even though the path integral approximation is very good.