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 stars 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.
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