Of the various homogenization approaches, the asymptotic expansion homogenization (AEH) approach for homogenizing nonlinear composite material properties continues to grow in prominence due to its ability to handle complex microstructural shapes while relating continuum fields of different scales. The objective is to study the AEH approach for nonlinear thermal heat conduction with temperature-dependent conductivity. First, two approaches are proposed to investigate the sensitivity of the homogenized conductivity to higher-order terms of the asymptotic series. Under conditions of symmetry such as in unidirectional composites, the two approaches give the same homogenized properties. Then validations are shown for unidirectional composites for changing volume fraction and temperature. The validations are performed using measurements and analytical formulas available in the literature. The findings show good agreement between the present numerical predictions and independent results. Finally, a simple nonlinear steady-state heat conduction problem is demonstrated to illustrate the multi-scale procedure. The numerically predicted results are verified using a Runge-Kutta solution.