A theory of thermal transport in a two-channel Kondo system, such as the one formed by a small quantum dot coupled to two leads and to a larger dot, is formulated. The interplay of the two screening constants allow an exploration of the Fermi liquid and non-Fermi liquid regimes. By using analytical, as well as numerical, renormalization group methods, we study the temperature dependence of the thermal conductance and the Lorentz number. We find that in the low-temperature limit, the Lorentz number attains its universal value, irrespective of the nature of the ground state.