We investigate a model for a rigid heat conductor which allows for variation of thermal properties at a microstructure level. With increasing use of nanofluids, nanomaterials, and the need to store materials at cryogenic temperatures we believe theories for bodies with (anisotropic) thermal microstructure will become increasingly important. In this article we examine how the solution depends on changes in coupling coefficients between the macro and microthermal level. A precise a priori continuous dependence result is established. The way in which the solution to the microstructural problem converges to that for a rigid body without thermal microstructure is also investigated.