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Recent observations have found several candidates for old warm neutron starswhose surface temperatures are above the prediction of the standard neutronstar cooling scenario, and thus require some heating mechanism. Motivated bythese observations, we study the non-equilibrium beta process in the minimalcooling scenario of neutron stars, which inevitably occurs in pulsars. Thisout-of-equilibrium process yields the late time heating in the core of aneutron star, called the rotochemical heating, and significantly changes thetime evolution of the neutron star surface temperature. To perform a realisticanalysis of this heating effect, we include the singlet proton and tripletneutron pairing gaps simultaneously in the calculation of the rate andemissivity of this process, where the dependence of these pairing gaps on thenucleon density is also taken into account. We then compare the predictedsurface temperature of neutron stars with the latest observational data. Weshow the simultaneous inclusion of both proton and neutron gaps is advantageousfor the explanation of the old warm neutron stars since it enhances the heatingeffect. It is then found that the observed surface temperatures of the old warmmillisecond pulsars, J2124-3358 and J0437-4715, are explained for variouschoices of nucleon gap models. The same setup is compatible with the observedtemperatures of ordinary pulsars including old warm ones, J0108-1431 andB0950+08, by choosing the initial rotational period of each neutron staraccordingly. In particular, the upper limit on the surface temperature ofJ2144-3933 can be satisfied if its initial period is $\gtrsim 10\,\mathrm{ms}$.

Cooling Theory Faced with Old Warm Neutron Stars: Role of Non-Equilibrium Processes with Proton and Neutron Gaps