The heat capacity of neutron matter is studied over the range of densities and temperatures prevailing in neutron-star crusts, allowing for the transition to a superfluid phase at temperatures below some critical temperature $T_{sf}$ and including the transition to the classical limit. Finite temperature Hartree-Fock-Bogoliubov equations (FTHFB) are solved and compared to existing approximate expressions. In particular, the formula given by Levenfish and Yakovlev is found to reproduce the numerical results with a high degree of accuracy for temperatures $T\leq T_{sf}$. In the non-superfluid phase, $T\geq T_{sf}$, the linear approximation is valid only at temperature $T\ll T_{{\rm F} n}$ ($T_{{\rm F} n}$ being the Fermi temperature of the neutron gas) which is rarely the case in the shallow layers of the neutron star's crust. A non-perturbative interpolation between the quantal and the classical regimes is proposed here. The heat capacity, conveniently parametrized solely in terms of $T_{sf}$, $T_{{\rm F} n}$, and the neutron number density $n_n$, can be easily implemented in neutron-star cooling simulations.