  Within the t-J model, the heat transport of the doped two-leg ladder material
Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$ is studied in the doped regime where
superconductivity appears under pressure in low temperatures. The energy
dependence of the thermal conductivity $\kappa_{c}(\omega)$ shows a low-energy
peak, while the temperature dependence of the thermal conductivity
$\kappa_{c}(T)$ is characterized by a broad band. In particular,
$\kappa_{c}(T)$ increases monotonously with increasing temperature at low
temperatures $T<0.05J$, and is weakly temperature dependent in the temperature
range $0.05J<T<0.1J$, then decreases for temperatures $T>0.1J$, in qualitative
agreement with experiments. Our result also shows that although both dressed
holons and spinons are responsible for the thermal conductivity, the
contribution from the dressed spinons dominates the heat transport of
Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$.
