  The physics of compact objects is one of the very actively developing
branches of theoretical investigations, since the careful analysis of different
models for the internal structure of these objects, their evolutionary
characteristics and the comparison with modern observational data give access
to discriminate among various speculations about the state of matter under
extreme conditions.
  The lecture provides an overview to the problem of neutron star cooling
evolution. We discuss the scheme and necessary inputs for neutron star cooling
simulations from the background of a microscopic modeling in order to present
the surface temperature - age characteristics of hybrid neutron stars including
the determination of main regulators of the cooling process, the question of
neutrino production and diffusion.
