  We study the thermal conductivity in disordered $s$-wave superconductors.
Expanding on previous works for normal metals, we develop a formalism that
tackles particle diffusion as well as the weak localization (WL) and weak
anti-localization (WAL) effects. Using a Green's functions diagrammatic
technique, which takes into account the superconducting nature of the system by
working in Nambu space, we identify the system's low-energy modes, the diffuson
and the Cooperon. The time scales that characterize the diffusive regime are
energy dependent; this is in contrast with the the normal state, where the
relevant time scale is the mean free time $\tau_e$, independent of energy. The
energy dependence introduces a novel energy scale $\varepsilon_*$, which in
disordered superconductors ($\tau_e \Delta\ll 1$, with $\Delta$ the gap) is
given by $\varepsilon_* = \sqrt{\Delta/\tau_e}$. From the diffusive behavior of
the low-energy modes, we obtain the WL correction to the thermal conductivity.
We give explicitly expressions in two dimensions. We determine the regimes in
which the correction depends explicitly on $\varepsilon_*$ and propose an
optimal regime to verify our results in an experiment.
