I will discuss a new mode of heat conduction in the neutron star inner crust. The collective excitations of the neutron superfluid (superfluid phonons) provide an efficient mode for heat conduction since they have relatively large mean free paths. I will describe the low-energy effective theory needed to calculate the superfluid phonon mean free path in the neutron star crust. This new mode of heat transport can be especially important in the thermal evolution of magnetars.