We study the effect of temperature in the crust on the style of faulting during the extension of a model lithosphere. Two distinct fault patterns are seen depending on whether the brittle layer thickness HL is greater or less than a critical value HC. For a thin layer (HL HC) a rift develops with multiple faults bounding an evolving set of grabens. Numerical results show that the value of HC depends on how the thermal structure and therefore the thickness distribution of the lithosphere evolve during extension. In the region where the displacement occurs along the fault the brittle lithosphere thins, and we show that this promotes development of multiple faults. In models with heat transfer only by rock motion and conduction, the advective thinning effect can greatly reduce HC. With rift velocities greater than 1 cm yr-1, HC is reduced by a factor of 5 (to unreasonably small values). However, inclusion of simulated hydrothermal cooling increases HC. This allows large offsets to develop in model lithospheres with brittle layers of order 10 km thick. Extension of layers thicker than 10 km tends to produce a rift with multiple graben structures. Finally, we compare the result of a simplified analytic model of extension to data. We find that the style of faulting seen in a variety of continental rifts is broadly consistent with our model predictions.