My balcony compost tends to freeze and stop working during winter time because I live in Quebec and we get an average of -13°C. Is there anything I can do to prolong my composting season?
Without some kind of external heating mechanism (eg geothermal spring, waste heat from a business), you'd need to rely on lots of insulation and the heat generated from the compost as it breaks down.
Possibly a long sheet of stainless steel bent into a U shape (when viewed from above) and insulated around the outside could help the compost keep its heat into winter, and still be durable enough to be worth the effort. In a very cold area such as yours, you'd probably want to add a lid, front and a base to the structure so that every side is insulated (including underneath), which could be very tricky but it may still be possible if you were really keen.
Gaps would need to be minimal, as a draft flowing through would contribute to cooling. I suggested a U shape so you can access the material easily to turn it, which helps keep up the level of heat. I've found that a largely air tight compost bin can do a great job as long as there's not too much wet green material, and in my region (warm and very dry) it's helpful to avoid air flow to prevent the material drying out.
Unfortunately large sheets of stainless steel may not be easy to obtain, but I guess a slightly damaged one might become available at a local metal recycler. After all, it is a common material used in manufacturing (eg appliances) and may not be usable for many purposes if creased by an accidental knock.
Because stainless steel lasts a very long time, you could consider it to be a sustainable solution, unlike a similar commercial design I saw called a "Hotter Rotter" which uses a PVC lining with soft foam insulation and a hard plastic outer shell.
I consider most types of insulation sold commercially to not be particularly sustainable, although wool could be an option if you are certain you can keep out the moisture. That could be addressed in a separate question anyway.
I'd be surprised though if even an insulated compost bin could avoid freezing all winter long in temperatures of -13C, but given enough compost and enough insulation it is certainly possible (though likely not cost effective).
Making use of a broken chest freezer might be an option (ensure it's been degassed), as the insulation is already taken care of. If you could protect the inside lining from damage somehow, then it could even last you quite some time.