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We introduce the theoretical basis for modeling the cooling evolution of compact stars starting from Boltzmann equations in curved space-time. We open a discussion on observational verification of different neutron star models by consistent statistics. Particular interest has the question of existence of quark matter deep inside of compact object, which has to have a specific influence on the cooling history of the star. Besides of consideration of several constraints and features of coolingdoi:10.1051/epjconf/20100703004 fatcat:mp7cm36atfgpxa6rzomj6dieta