Selected problems in astrophysics of compact objects

I review three problems in astrophysics of compacts stars: (i) the phase diagram of warm pair-correlated nuclear matter a sub-saturation densities at finite isospin asymmtery; (ii) the Standard Model neutrino emission from superfluid phases in neutron stars within the Landau theory of Fermi (superfluid) liquids; (iii) the beyond Standard Model physics of axionic cooling of compact stars by the Cooper pair-breaking processes.