  A critical relation in the study of neutron star cooling is the one between
surface temperature and interior temperature. This relation is determined by
the composition of the neutron star envelope and can be affected by the process
of diffusive nuclear burning (DNB), which occurs when elements diffuse to
depths where the density and temperature are sufficiently high to ignite
nuclear burning. We calculate models of H-He and He-C envelopes that include
DNB and obtain analytic temperature relations that can be used in neutron star
cooling simulations. We find that DNB can lead to a rapidly changing envelope
composition and prevents the build-up of thermally stable hydrogen columns
y$_H$ > 10$^{7}$ g cm$^{-2}$, while DNB can make helium envelopes more
transparent to heat flux for surface temperatures $T_s$ > 2 $\times 10^6$ K. We
perform neutron star cooling simulations in which we evolve temperature and
envelope composition, with the latter due to DNB and accretion from the
interstellar medium. We find that a time-dependent envelope composition can be
relevant for understanding the long-term cooling behaviour of isolated neutron
stars. We also report on the latest Chandra observations of the young neutron
star in the Cassiopeia A supernova remnant; the resulting 13 temperature
measurements over more than 18 years yield a ten-year cooling rate of $\approx$
2%. Finally, we fit the observed cooling trend of the Cassiopeia A neutron star
with a model that includes DNB in the envelope.
