it is difficult to answer this question generally for all cases. But it can be well answered in the cases you addressed.
In most cases an anisotrop crystal symmetry is responsible for anisotrop properties.
-In the case of thermal conductivity k the anisotrop velocity of sound vS is responsible for an anisotrop k. (k = vS*c*l/3 with c the specific heat and l the mean free path). The anisotropy in vS is due to an anisotrop Young modul and this is the case for anisotrop crystal structures. As an example, ß-Ga2O3 has an monoclin structure and a vS which depends on the crystal direction. Therefrom, a anisotrop k follows.
-The energy gap is an minimum energy difference between 2 different points in the momentum space. Therefore, the gap cannot be anisotrop, because it is no property which depends on a direction.
Thank you Dr. Halenkovic and Dr. Mitdank for your inputs.
Dr. Mitdank, it is interesting that you mention the "gap cannot be anisotropic". I have seen various reports of different absorption edges along different axes in Ga2O3 using polarized optical absorption and reflectance measurements of Ga2O3. For instance,
Article Valence band ordering in β-Ga 2 O 3 studied by polarized tra...
summarizes them. Can I now ask naively that how are band gap and the absorption edges in a polarized optical absorption different?