We propose a heat valve based on the interplay between thermal transport and proximity-induced exchange splitting in Josephson tunnel junctions. We demonstrate that the junction electron heat conductance strongly depends on the relative alignment of the exchange fields induced in the superconductors. Colossal magnetothermal resistance ratios as large as are predicted to occur under proper temperature and phase conditions, as well as suitable ferromagnet-superconductor combinations. Moreover, the quantum phase tailoring, intrinsic to the Josephson coupling, offers an additional degree of freedom for the control of the heat conductance. Our predictions for the phase-coherent and spin-dependent tuning of the thermal flux can provide a useful tool for heat management at the nanoscale.
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