  We compute the contribution of phonons to the thermal conductivity in the
core of superfluid neutron stars. We use effective field theory techniques to
extract the phonon scattering rates, written as a function of the equation of
state of the system. We also calculate the phonon dispersion law beyond linear
order, which depends on the gap of superfluid neutron matter. With all these
ingredients, we solve the Boltzmann equation numerically using a variational
approach. We find that the thermal conductivity $\kappa$ is dominated by
combined small and large angle binary collisions. As in the color-flavor-locked
superfluid, we find that our result can be well approximated by $\kappa \propto
1/ \Delta^6$, where $\Delta$ is the neutron gap, the constant of
proportionality depending on the density. We further comment on the possible
relevance of electron and superfluid phonon collisions in obtaining the total
contribution to the thermal conductivity in the core of superfluid neutron
stars.

Solar and Stellar Astrophysics;High Energy Physics - Phenomenology;Nuclear Theory