Anisotropic thermal transport induced by deformation and the linear relation between the thermal conductivity and stress tensors, also known as the stress-thermal rule (STR), are tested via molecular dynamics simulations in well-entangled linear polyethylene (PE) and polystyrene (PS) melts subjected to extensional flow. We propose a method to determine the stress in deformed molecular melts, a key component missing in prior simulation studies on thermal transport in polymers that prevented verification of the STR. We compare our results with available data from previous experimental and simulation studies. Thermal conductivity (TC) is found to increase (decrease) in the direction parallel (perpendicular) to the imposed stretch. We find that the STR is valid for both PE and PS over a wide range of deformation rates and stress levels. In direct agreement with experimental evidence and the STR, we observe that for a given strain, the anisotropy in TC increases with the strain rate. Surprisingly, our results for PE question the universal behavior with respect to polymer chemistry suggested by experiments by showing a significantly higher proportionality constant (the stress-thermal coefficient) between stress and anisotropy in TC. We argue that this discrepancy can be explained by the high degree of entanglement interactions in PE affecting the transport of energy at the molecular level. Our conjecture is tested by studying an entangled linear PS melt, a polymer with a much lower entanglement plateau, for which thermal transport experimental results are available. For PS, the normalized stress-thermal coefficient is found to be commensurate with the experimental value. Finally, we test the fundamental molecular hypothesis of preferential energy transport along the backbone of polymer chains used to formulate the STR, which was prompted by early experimental evidence showing an increase in TC with chain length. We are able to establish that the increase in TC with chain length in PE melts fades as the system becomes entangled (i.e., TC remains constant beyond the critical entanglement chain length that marks the transition to entanglement-dominated rheological behavior). Our findings are of key importance in developing robust molecular-to-continuum methodologies for the study of nonisothermal macroscopic flows that are extremely relevant to polymer manufacturing processes.