  Valuable information about the neutron star interior can be obtained by
comparing observations of thermal radiation from a cooling neutron star crust
with theoretical models. Nuclear burning of lighter elements that diffuse to
deeper layers of the envelope can alter the relation between surface and
interior temperatures and can change the chemical composition over time. We
calculate new temperature relations and consider two effects of diffusive
nuclear burning (DNB) for H-C envelopes. First, we consider the effect of a
changing envelope composition and find that hydrogen is consumed on short
timescales and our temperature evolution simulations correspond to those of a
hydrogen-poor envelope within ~100 days. The transition from a hydrogen-rich to
a hydrogen-poor envelope is potentially observable in accreting NS systems as
an additional initial decline in surface temperature at early times after the
outburst. Second, we find that DNB can produce a non-negligible heat flux, such
that the total luminosity can be dominated by DNB in the envelope rather than
heat from the deep interior. However, without continual accretion, heating by
DNB in H-C envelopes is only relevant for <1-80 days after the end of an
accretion outburst, as the amount of light elements is rapidly depleted.
Comparison to crust cooling data shows that DNB does not remove the need for an
additional shallow heating source. We conclude that solving the time-dependent
equations of the burning region in the envelope self-consistently in thermal
evolution models instead of using static temperature relations would be
valuable in future cooling studies.
