The thermal conductivity (lambda) of stretched amorphous atactic polystyrene (PS) swollen in supercritical carbon dioxide (sc CO2) has been investigated over a wide temperature, pressure, and concentration range. Nonequilibrium molecular dynamics simulations with a full atomistic force-field have been employed to calculate the thermal conductivity of neat stretched PS and of different mixtures of supercritical CO2 with stretched PS. As the energy transport in PS parallel and perpendicular to the stretching direction differs, an anisotropy in the thermal conductivity occurs. The magnitude of lambda is enhanced with an increasing number of carbon-carbon backbone bonds oriented parallel to the direction of the heat transport. The degrees of freedom in the side chain of the polymer are rather unimportant for the thermal conductivity. To understand the conditions leading either to an equivalence or nonequivalence of the system degrees of freedom for the heat transport, we have analyzed lambda of PS, CO2, binary PS-CO2 mixtures and other model systems as a function of the bond constraints in the computational model. Furthermore, we have commented on differences in the thermal conductivity provided either by a vibrational energy transfer or by collisions.