  The thermal conductivity of the color-flavor locked phase of dense quark
matter is calculated. The dominant contribution to the conductivity comes from
photons and Nambu-Goldstone bosons associated with breaking of baryon number
which are trapped in the quark core. Because of their very large mean free path
the conductivity is also very large. The cooling of the quark core arises
mostly from the heat flux across the surface of direct contact with the nuclear
matter. As the thermal conductivity of the neighboring layer is also high, the
whole interior of the star should be nearly isothermal. Our results imply that
the cooling time of compact stars with color-flavor locked quark cores is
similar to that of ordinary neutron stars.
