  The study of how neutron stars cool over time can provide invaluable insights
into fundamental physics such as the nuclear equation of state and
superconductivity and superfluidity. A critical relation in neutron star
cooling is the one between observed surface temperature and interior
temperature. This relation is determined by the composition of the neutron star
envelope and can be influenced by the process of diffusive nuclear burning
(DNB). We calculate models of envelopes that include DNB and find that DNB can
lead to a rapidly changing envelope composition which can be relevant for
understanding the long-term cooling behavior of neutron stars. We also report
on analysis of the latest temperature measurements of the young neutron star in
the Cassiopeia A supernova remnant. The 13 Chandra observations over 18 years
show that the neutron star's temperature is decreasing at a rate of 2-3 percent
per decade, and this rapid cooling can be explained by the presence of a proton
superconductor and neutron superfluid in the core of the star.
