The presence of a neutron superflow in the inner crust and outer core of neutron stars can lead to a regime in which neutrons remain superfluid even though the energy spectrum of quasiparticle excitations exhibits no gap. For such gapless superfluidity, we have shown within the nuclear energy density functional theory that the neutron specific heat is comparable to that in the normal phase. We have studied the implication of gapless superfluidity for the thermal relaxation of transiently accreting neutron stars and compared our calculations with observations.