The Inuit seem to have used bows of wood, bone or antler backed with sinew cabling. These cordage backed bows probably count as a form of composite. Obviously such a weapon would have to work fairly well in the cold to be worth the considerable bother of making. This suggests that the components of a composite retain their material properties in serious cold. This page gives some details.
I don't know how extreme cold would effect the glue for holding the layers together - which in temperate climates can take on the order of two years to set fully. I could see being repeatedly frozen during that time causing issues, but really have no hard evidence. Since a cordage backed bow can be made entirely with knotwork, unfortunately the Inuit designs just don't offer any insight here.