erm... blush... that may be where I got it from too... lemme check...
The specific heat of a superconductor is not zero, but is much smaller than what a normal metal would have.
At low temperatures, the specific heat of a normal metal goes as c = A·T + B·T^3 (the linear term being conduction by electrons and the cubic term by phonons, i.e., lattice vibrations). If we drop the temperature below Tc (the critical temperature for the superconducting phase change), c increases and then falls much more rapidly than what the previous formula indicates (exponentially). If we now apply a strong magnetic field (Bc), the superconductivity is destroyed
[*] and we can measure the specific heat of the normal metal at that temperature. It is much higher.
The precise form of the superconducting heat capacity can, on ocassion, be explained by an energy gap. This is consistent with the BCS theory of superconductivity. Keep in mind, however, that there are many kinds of superconductors and this theory does not work for all of them.
If you want some rough explanation of why superconductors have small specific heats, you can think of them as being more ordered (lower entropy).
Edited to add:
Ouch! I now realise you were talking about thermal conductivity, not specific heat. Superconductors, contrary to what one would think, are poor thermal conductors and don't exhibit the Peltier effect. This indicates that the conducting electrons don't carry entropy. As before, we can measure thermal conductivity at a given temperature for the normal and superconducting states by application of a magnetic field and we see that the normal metal is a better heat conductor. This poor conductivity makes them useful for making thermal switches.
All this seems contrary to intuition: we have learned that good electrical conductors are also good conductors of heat (after all, the electrons carry both current and heat). But this doesn't work with superconductors, because in them the electrons are not independent, but coupled.
________________
[*] At very low temperatures and zero magnetic field, some metals are superconductors. If we apply a sufficiently strong magnetic field they become normal conductors again. At each temperature there is a critical magnetic field, Bc. At the critical temperature for the superconducting phase transition this critical field is zero. With some materials (Type I superconductors), the change is very sudden. Type II superconductors experiment a transition state (vortex state) and have two critical fields. This is quite important for the practical applications of superconductors (magnetic resonance machines in hospitals use superconductors to generate their magnetic fields, if those superconductors had a small Bc they would be worthless).