Check what all in its daily life is dusty (hay, grain, etc). When I was having this problem, we changed my boy from sawdust/shaving mix bedding to pure shavings, started soaking his hat and his grain. The problem seemed to clear up in a week or so.
If the temperature just started to drop, it's possible it's just his reaction to finding it more difficult to breathe..
Soaking hay is good, though when my horse was coughing it was only due to a loose tooth..
You can get as Melinda said oxygen products from Ravene. That's only plants so you don't risk anything. You can also work outside!
First of all, you should wet the hay, that's maybe because of the dust that your horse is coughing. Raven's products are pretty good if it doesn't because of the temperature you should think about it :)