What is causing discoloration on my bronze-finished lockset?
It's called a tarnish or patina and is caused by oxidation of the copper in the bronze.  The sun's heat will accelerate that reaction somewhat compared to one in a cooler place.

If you don't like it, simply clean the lockset with Brasso when it starts to look discolored.  It will take some elbow grease the first time because the tarnish has had time to build up, but if you keep up with it, subsequent times will be a lot easier.