Is TMJ pain exacerbated by hot and cold?

More cold than hot. Cold weather often makes TMJ pain worse. Heat, not usually.
Maybe. Tmj pain when related to an inflammation in the joint is usually exacerbated by heat and alleviated by cold. However if you have a neuromuscular disorder in one muscle next to the TMJ ( lateral pterygoid) or in one muscle that refers pain to the TMJ area, heat usually alleviates the pain while cold can numb the area but afterwards may increase the pain due to muscle contraction.
Possible. Anything is possible when you already have TMJ pain. Get evaluated by a dentist to identify the cause of the TMJ pain ( most of my patients have a long history of night time grinding or clenching of teeth, without realizing it), and get it corrected( most likely with a special dentist-fabricated night guard).
Sometimes. Normally, tmj/tmd pain, if associated with a neuromuscular component, can react in different ways to temperature change. The facial muscles and the muscles that are part of the mandibular complex may be in spasm, and heat or cold can cause reactions. I would see a tmj/tmd specialist to discuss and evaluate your issues.