Sorry to hear it stings so much when you use it at room temperature. How long have you been using Copaxone? Just keep in mind what the med is actually doing as it injects into your system. Breaking down fatty tissue AND attaching itself to your DNA strands, to make you well.
I compare the discomfort to getting a vaccine. Are you taking tylenol or ibuprofin before you take you shots?
I can't answer about warming up the area with a hot pack, I don't bother. Best of Luck with your shots