  The thermal evolution of neutron stars is coupled to their spin down and the
resulting changes in structure and chemical composition. This coupling
correlates stellar surface temperatures with rotational state as well as time.
We report an extensive investigation of the coupling between spin down and
cooling for hybrid stars which undergo a phase transition to deconfined quark
matter at the high densities present in stars at low rotation frequencies. The
thermal balance of neutron stars is re-analyzed to incorporate phase
transitions and the related latent heat self-consistently, and numerical
calculations are undertaken to simultaneously evolve the stellar structure and
temperature distribution. We find that the changes in stellar structure and
chemical composition with the introduction of a pure quark matter phase in the
core delay the cooling and produce a period of increasing surface temperature
for strongly superfluid stars of strong and intermediate magnetic field
strength. The latent heat of deconfinement is found to reinforce this signature
if quark matter is superfluid and it can dominate the thermal balance during
the formation of a pure quark matter core. At other times it is less important
and does not significantly change the thermal evolution.
