\[to $\sim (10-20)\rho_0$ in most massive stars\]. Its composition is very uncertain. It may be the same as in the outer core or essentially different. In particular, hyperons may appear there in addition to nucleons. Another possibility is the appearance of exotic matter (pion or kaon condensates or strange quark matter or mixed phases, as reviewed, e.g., in [@nsb1]). Nucleons, hyperons and quarks can be in superfluid state. Physical properties of matter at $\rho \la \rho_0$ are more or less restricted by nuclear physics data, but at higher $\rho$ they are uncertain. Typical baryon chemical potentials $\sim 500$ MeV in a neutron star core are most difficult for rigorous microscopic calculations. Note a lively discussion in the literature that compact stars (or some of them) can be not neutron stars but strange stars (e.g., Refs. [@glendenning96; @nsb1; @lp07]) built entirely or almost entirely of strange quark matter.

Neutron stars are born in supernova explosions with high internal temperature $T \sim 10^{11}$ K, but gradually cool down. In $\sim$30 s after the birth a star becomes transparent for neutrinos generated in its interiors and transforms from a proto-neutron star (e.g., [@ponsetal01]) to an ordinary neutron star whose EOS is almost temperature 