  We discuss thermal transport through a Josephson junction in a time-dependent
situation. We write the spectral representation of the heat current pumped by a
generic drive. This enables separation of the dissipative and reactive
contributions, of which the latter do not contribute to long-time averages. We
discuss the physical interpretation, and note that the condensate heat current
identified in [K. Maki and A. Griffin, Phys. Rev. Lett. 15, 921 (1965)] is
purely reactive. The results enable a convenient description of heat exchanges
in a Josephson system in the presence of an external drive, with possible
applications for the implementation of new cooling devices.
