I've noticed that the AC in my CX-5 pumps out cold air only if I have the temperature setting at 68 deg. or less (if it's hot out I set it to 66 deg until the car cools off, then reset to the actual desired temp). If it's above that point, the air that comes out is "somewhat cool" but not cold. When I set it for cold air, I'd say the cold air is almost immediate (within 1 minute). Of course I'm not subject to Az temps, and when it's sunny out I do use a reflective sun shade.
I live in the bay area and always have my climate control set between 70-73. I've never had an issue with the unit providing air that was too hot or too cold. I spent a good part of Saturday driving around and noticed that outside temps were reading from 80-100 degrees. I had my climate control set to 71 the whole time and was surprised when I saw the outside temp was 99 while sitting at the In-N-Out drive through, where the guy was outside taking orders. I put my hand to the windshield and moon roof just to see what it felt like, and it was HOT! The air blowing out never felt ice cold, but it also didn't need to be because inside the cabin I was comfortable at what I'm assuming was 71 degrees. I drive a 2015, so not sure if that has anything to do with it.