We have extracted temperature-dependent thermal conductivity values from scanning thermal microscopy measurements of a self-heated multiwalled carbon nanotube supported on a silicon substrate. A deliberately introduced segment of amorphous carbon served as an integrated nanoheater. Kelvin probe force microscopy was used to supplement the thermometry data with values for the nanotube's electrical resistivity. This way, both the spatially resolved temperature rise and the Joule heating power density were available for further analysis. A one-dimensional heat diffusion model was fitted to the data to extract values for the thermal conductivity along the nanotube axis and the thermal conductance between the nanotube and supporting substrate. We found thermal conductivity values that continuously increase from 200 to 400 W m-1 K-1 in a temperature range of 100 to 400 K above room temperature. The values obtained are about one order of magnitude lower compared to values reported for the freely suspended case. We attribute this observation to the increased phonon scattering and quenching of acoustic phonon modes due to the substrate interaction.