Comment by pfdietz

One notable exception to this is superconductors. One might naively think that because superconductors have zero electrical resistance, they also have zero thermal resistance. But this is wrong (sorry, Larry Niven)! The superconducting charge carriers (Cooper Pairs) have zero entropy, so they can't carry heat. Thermal conductivity of a superconducting material drops when it becomes superconductive.

I believe high Tc superconductors have been used (or at least proposed to be used) as current leads for carrying current into low Tc superconductors from somewhat higher temperature normal conductors.