I can't see the bad area, so I can't really say if it is rot. 40F is not cold for this plant, it is comfortably hardy to freezing and below. 40F is a good temperature to keep it at all winter, but it should be dry. Wet soil at these cool temperatures is likely to cause rotting. A plant that starts to rot will die fairly quickly. Sometimes a small area of rot can be cut out and the plant will recover, but often it starts in the roots and the whole of the inside is liquid before you even see a mark on the outside. Rotten areas will be very very soft.

I'd be tempted to unpot it, gently remove some soil, and any rotted areas, then leave it to dry out completely. You can repot in the spring, using a largely inorganic soil mix (perlite, grit, fired clay, pumice, coarse sand). Avoid using peat moss.