  We discuss how phase coherence manifests in the heat transport through
superconducting single and multichannel Josephson junctions in time dependent
situations. We consider the heat current driven through the junction by a
temperature difference in dc voltage and ac phase biased situations. At low
bias, the electromagnetic driving mainly modifies the form of the coherent
resonance that transports a large part of the heat current, rather than simply
dissipating energy in the junction. We find a description for the heat current
in terms of quasiparticle n-photon absorption and emission rates, and discuss
analytical and numerical results concerning them. In addition to the ensemble
average heat transport, we describe also its fluctuations.
