I have asthma and live in Colorado so I know cold. My exercise room is a bit cool also, probably not as cold as yours.
First I usually start off with a flannel shirt that is easy to take off while I'm moving once I've warmed up.
I know outside I use a scarf around my mouth and nose so that the air I take in is warmed up. If a scarf is too warm maybe one of those regular paper type face masks until you get going and warmed up.
I'm sure someone who actually knows what they are talking about will be by shortly.
Karen