Understanding the thermal evolution of sedimentary basins is crucial for the geosciences and petroleum genesis. A recent model of thermal conductivity, based on microscopic physics, has shown that the variations of thermal conductivity depend nonlinearly on the temperature, pressure and mineralogy. The time-dependent temperature equation takes on a nonlinear diffusive character with the temperature-dependent portion of the thermal conductivity. We have compared the numerical solutions of the temperature fields, following a lithospheric stretching event between constant thermal conductivity and variable thermal conductivity. There is a great sensitivity of the thermal signatures to the parameter governing the influence of mineralogy on the phonon contribution to the thermal conductivity. For an olivine-like lithosphere, differences on the order of 5% or 60 to 70K in the interior temperature between the two models can be expected. For times younger than 150 Myr, the topography for the variable conductivity model is shallower by around 5 to 10% than for the constant conductivity model, while the surface heat flow remains always lower for variable conductivity and the departure between the two conductivity models can reach 20% in the early stage. Variable thermal conductivity may play a non-trivial role in thermal-chemical reactive processes under basins.