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 Greens functions diagrammatic technique, which takes into account the superconducting nature of the system by working in Nambu space, we identify the systems 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 ($tau_e Deltall 1$, 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.
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