The double-layer contribution to the single-particle thermal diffusion coefficient of charged, spherical colloids with arbitrary double-layer thickness is calculated and compared to experiments. The calculation is based on an extension of the Debye-Hückel theory for the double-layer structure that includes a small temperature gradient. There are three forces that constitute the total thermophoretic force on a charged colloidal sphere due to the presence of its double layer: i) the force F W that results from the temperature dependence of the internal electrostatic energy W of the double layer, ii) the electric force F el with which the temperature-induced non-spherically symmetric double-layer potential acts on the surface charges of the colloidal sphere and iii) the solvent-friction force F sol on the surface of the colloidal sphere due to the solvent flow that is induced in the double layer because of its asymmetry. The force F W will be shown to reproduce predictions based on irreversible-thermodynamics considerations. The other two forces F el and F sol depend on the details of the temperature-gradient-induced asymmetry of the double-layer structure which cannot be included in an irreversible-thermodynamics treatment. Explicit expressions for the thermal diffusion coefficient are derived for arbitrary double-layer thickness, which complement the irreversible-thermodynamics result through the inclusion of the thermophoretic velocity resulting from the electric- and solvent-friction force.