We calculate the neutrino mean free path in color superconducting quark mat
ter, and employ it to study the cooling of matter via neutrino diffusion in
the superconducting phase as compared to a free quark phase. The cooling p
rocess slows when quark matter undergoes a second order phase transition to
a superconducting phase at the critical temperature T-c. Cooling subsequen
tly accelerates as the temperature decreases below T-c. This will directly
impact the early evolution of a newly born neutron star, should its core co
ntain quark matter. Consequently, there may be observable changes in the ea
rly neutrino emission which would provide evidence for superconductivity in
hot and dense matter. [S0556-2821(00)07320-3].