  The thermodynamics of pairing phase-transition in nuclei is studied in the
canonical ensemble and treating the pairing correlations in a
finite-temperature variation after projection BCS approach (FT-VAP). Due to the
restoration of particle number conservation, the pairing gap and the specific
heat calculated in the FT-VAP approach vary smoothly with the temperature,
indicating a gradual transition from the superfluid to the normal phase, as
expected in finite systems. We have checked that the predictions of the FT-VAP
approach are very accurate when compared to the results obtained by an exact
diagonalization of the pairing Hamiltonian. The influence of pairing
correlations on specific heat is analysed for the isotopes $^{161,162}$Dy and
$^{171,172}$Yb. It is shown that the FT-VAP approach, applied with a level
density provided by mean field calculations and supplemented, at high energies,
by the level density of the back-shifted Fermi gas model, can approximate
reasonably well the main properties of specific heat extracted from
experimental data. However, the detailed shape of the calculated specific heat
is rather sensitive to the assumption made for the mean field.
