  We formulate the finite-temperature relativistic Hartree-Bogoliubov theory
for spherical nuclei based on a point-coupling functional, with the Gogny or
separable pairing force. Using the functional PC-PK1, the framework is applied
to the study of pairing transitions in Ca, Ni, Sn, and Pb isotopic chains. The
separable pairing force reproduces the gaps calculated with the Gogny force not
only at zero temperature, but also at finite temperatures. By performing a
systematic calculation of the even-even Ca, Ni, Sn, and Pb isotopes, it is
found that the critical temperature for a pairing transition generally follows
the rule $T_c =0.6 \Delta_n(0)$, where $\Delta_n(0)$ is the neutron pairing gap
at zero temperature. This rule is further verified by adjusting the pairing gap
at zero temperature with a strength parameter.
