I vastly prefer running in the evenings, for cooling reasons - at a trot I put out more BTUs than my roommate's brazier, I kid you not.

Well, okay, that's a bit of an exaggeration. But not by much!
Most people run either in the early am or the evening because it's cooler. Were I you, I would run in the afternoon. Just realize it'll be a bit more difficult, because of the heat, and you will need to take more care with your hydration. Don't recommend waking up, running and then going back to sleep. It might mess with your (circadian) rhythm, and likely lessen the quality of your sleep (due to all those pump-you-up exercise endorphins swimming about in you).