The finite-temperature variation after projection BCS approach (FT-VAP) is employed to study the pairing phase-transition for the 161, 162Dy and 171,172Yb isotopes. 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. FT-VAP results are compared with those obtained in the exact canonical approach as well as with experimental data.