I have a similar reaction. Cool air below 40 or so always bothers me a little bit when breathing very heavy when climbing hills and such on the bike.

I cough a little bit for a day or so after a long ride in that kind of air. Maybe a lot of is has to do with the dryness of the air as much as it does the temp. Aside from the cough, it doesn't seem to be a serious thing.

Drink more often on your ride when you notice this. That seems to help my issues some.