  The properties of superdense matter in neutron star (NS) cores control NS
thermal states by affecting the efficiency of neutrino emission from NS
interiors. To probe these properties we confront the theory of thermal
evolution of NSs with observations of their thermal radiation. Our
observational basis includes cooling isolated NSs (INSs) and NSs in quiescent
states of soft X-ray transients (SXTs). We find that the data on SXTs support
the conclusions obtained from the analysis of INSs: strong proton superfluidity
with T_{cp,max} >= 10^9 K should be present, while mild neutron superfluidity
with T_{cn,max} =(2*10^8 -- 2*10^9) K is ruled out in the outer NS core. Here
T_{cn,max} and T_{cp,max} are the maximum values of the density dependent
critical temperatures of neutrons and protons. The data on SXTs suggest also
that: (i) cooling of massive NSs is enhanced by neutrino emission more powerful
than the emission due to Cooper pairing of neutrons; (ii) mild neutron
superfluidity, if available, might be present only in inner cores of massive
NSs. In the latter case SXTs would exhibit dichotomy, i.e. very similar SXTs
may evolve to very different thermal states.
