I've been hunting around and found a few items worth mentioning, in case anyone else in a similar situation:

* Doha Zoo. Opening hours are 7:30-11:30am and 3:30-8:30pm daily except Friday mornings. Phone 468 2610.
* Breakfast by the Hotel Pools or Beach. Open early for the executives, but very accomidating to the morning adventurer. You'll practically have the whole place to yourself.
* The Corniche. Not too hot to walk around early in the morning. Gives you a chance to enjoy the dhows setting out for the day, without having to worry about rushing back to the air conditioning.
* Marketplaces. You can get a jump on some of the open-air marketplaces.
* Desert Adventures. Can get into a morning tour of the sand dunes, or go down Salwa looking for desert rock formations. Getting out in the morning lets you beat the hot afternoon heat.