Astronomy meets QCD: cooling constraints for the theories of internal structure of compact objects

We discuss a set of tests which confront observations of cooling compact objects and theories of their thermal evolution. As an example we apply the recently developed $\mathrm{Log N}$-$\mathrm{Log S}$ test of compact star cooling theories to hybrid stars with a color superconducting quark matter core, we also apply and discuss other existing tests. While there is not yet a microscopically founded superconducting quark matter phase which would fulfill constraints from cooling phenomenology, we explore the hypothetical 2SC+X phase and show that the magnitude and density-dependence of the X-gap can be chosen to satisfy a set of tests: the temperature -- age ($\mathrm{T}$-$\mathrm{t}$) test, the $\mathrm{Log N}$-$\mathrm{Log S}$ test, the brightness constraint, and the mass spectrum constraint. Some recent modifications of the population synthesis model used to obtain the $\mathrm{Log N}$-$\mathrm{Log S}$ distribution are briefly discussed. In addition, we propose to use the age-distance diagram as a new tool to study the local population of young isolated neutron stars.