We study the consequences of superconducting quark cores ( with color-flavor-locked phase as representative example ) for evolution of temperature profiles and the cooling curves in quark-hadron hybrid stars and in hypothetical self-bounded objects having no a hadron shell ( quark core neutron stars ) .
The quark gaps are varied from 0 to \Delta _ { q } = 50 MeV .
For hybrid stars we find time scales of 1 \div 5 , 5 \div 10 and 50 \div 100 years for the formation of a quasistationary temperature distribution in the cases \Delta _ { q } = 0 , 0.1 MeV and \stackrel { \scriptstyle > } { \phantom { } { } _ { \sim } } 1 MeV , respectively .
These time scales are governed by the heat transport within quark cores for large diquark gaps ( \Delta \stackrel { \scriptstyle > } { \phantom { } { } _ { \sim } } 1 MeV ) and within the hadron shell for small diquark gaps ( \Delta \stackrel { \scriptstyle < } { \phantom { } { } _ { \sim } } 0.1 MeV ) .
For quark core neutron stars we find a time scale \simeq 300 years for the formation of a quasistationary temperature distribution in the case \Delta \stackrel { \scriptstyle > } { \phantom { } { } _ { \sim } } 10 MeV and a very short one for \Delta \stackrel { \scriptstyle < } { \phantom { } { } _ { \sim } } 1 MeV .
If hot young compact objects will be observed they can be interpreted as manifestation of large gap color superconductivity .
Depending on the size of the pairing gaps , the compact star takes different paths in the \mbox { lg } ( T _ { s } ) vs . \mbox { lg } ( t ) diagram where T _ { s } is the surface temperature .
Compared to the corresponding hadronic model which well fits existing data the model for the hybrid neutron star ( with a large diquark gap ) shows too fast cooling .
The same conclusion can be drawn for the corresponding self-bound objects .