  Recent experiments on imbalanced fermion gases have proved the existence of a
sharp interface between a superfluid and a normal phase. We show that, at the
lowest experimental temperatures, a temperature difference between N and SF
phase can appear as a consequence of the blocking of energy transfer across the
interface. Such blocking is a consequence of the existence of a SF gap, which
causes low-energy normal particles to be reflected from the N-SF interface. Our
quantitative analysis is based on the Hartree-Fock-Bogoliubov-de Gennes
formalism, which allows us to give analytical expressions for the thermodynamic
properties and characterize the possible interface scattering regimes,
including the case of unequal masses. Our central result is that the thermal
conductivity is exponentially small at the lowest experimental temperatures.
