  We calculate the electronic contribution to the thermal conductance in a
quantum dot that is weakly coupled via tunnel barriers to two electrons
reservoirs. A linear response model is derived for the calculation of the heat
current Q through the quantum dot when a small temperature difference and a
small voltage difference are applied between the two reservoirs. We find that
the thermal conductance oscillates as a function of the Fermi energy. The
periodicity of these oscillations is the same as this of the Coulomb blockade
oscillations of the conductance. In the quantum limit the peak values of the
thermal conductance tends to zero as exp(-DE/kT) where DE is the spacing
between the energy levels in the dot. This surprising result implies the
failure of the Wiedemann-Franz law in the quantum limit and it could lead to
structures with large figure of merit ZT. We have also examined the behavior of
the thermal conductance in the classical limit and we find that the
Wiedemann-Franz is recovered.
