  The interior of mature neutron stars is expected to contain superfluid
neutrons and superconducting protons. The influence of temperature and currents
on superfluid properties is studied within the self-consistent time-dependent
nuclear energy-density functional theory. We find that this theory predicts the
existence of a regime in which nucleons are superfluid (the order parameter
remains finite) even though the energy spectrum of quasiparticle excitations
exhibits no gap. We show that the disappearance of the gap leads to a specific
heat that is not exponentially suppressed at low temperatures as in the BCS
regime but can be comparable to that in the normal phase. Introducing some
dimensionless effective superfluid velocity, we show that the behavior of the
specific heat is essentially universal and we derive general approximate
analytical formulas for applications to neutron-star cooling simulations.
