Context. The traditional study of neutron star cooling has been generally applied to quite old objects such as the Crab Pulsar (957?years) or the central compact object in Cassiopeia A (330 years) with an observed surface temperature ?~106 K. However, recent observations of the late (t?=?108–109 s) emission of the supernovae (SNe) associated to GRBs (GRB-SN) show a distinctive emission in the X-ray regime consistent with temperatures ?~107–108 K. Similar features have been also observed in two Type Ic SNe SN?2002ap and SN 1994I that are not associated to GRBs. Aims. We advance the possibility that the late X-ray emission observed in GRB-SN and in isolated SN is associated to a hot neutron star just formed in the SN event, here defined as a neo-neutron star. Methods. We discuss the thermal evolution of neo-neutron stars in the age regime that spans from ?~1?min (just after the proto-neutron star phase) all the way up to ages ?<10–100 yr. We examine critically the key factor governing the neo-neutron star cooling with special emphasis on the neutrino emission. We introduce a phenomenological heating source, as well as new boundary conditions, in order to mimic the high temperature of the atmosphere for young neutron stars. In this way we match the neo-neutron star luminosity to the observed late X-ray emission of the GRB-SN events: URCA-1 in GRB980425-SN1998bw, URCA-2 in GRB030329-SN2003dh, and URCA-3 in GRB031203-SN2003lw. Results. We identify the major role played by the neutrino emissivity in the thermal evolution of neo-neutron stars. By calibrating our additional heating source at early times to ?~1012–1015 erg/g/s, we find a striking agreement of the luminosity obtained from the cooling of a neo-neutron stars with the prolonged (t?=?108–109 s) X-ray emission observed in GRB associated with SN. It is therefore appropriate a revision of the boundary conditions usually used in the thermal cooling theory of neutron stars, to match the proper conditions of the atmosphere at young ages. The traditional thermal processes taking place in the crust might be enhanced by the extreme high-temperature conditions of a neo-neutron star. Additional heating processes that are still not studied within this context, such as e+e? pair creation by overcritical fields, nuclear fusion, and fission energy release, might also take place under such conditions and deserve further analysis. Conclusions. Observation of GRB-SN has shown the possibility of witnessing the thermal evolution of neo-neutron stars. A new campaign of dedicated observations is recommended both of GRB-SN and of isolated Type Ic SN.
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