the roof is like a giant solar panel, as is the surrounding ground. sucking in hot air, may not prove to drop temps. if one shaded the roof, as well as the south side with some fabric, you can get some pretty dramatic results. i insulated the south side of my barn and the heat gain drop is about 7 degrees from before on those walls. i put a shade over the patio and the back of the garage which was acting as a heat sink. after installation, about $200, the temp in the house dropped 8 degrees from before if the outside temp didn't get over about 95. then it just raised linearly. we don't have air conditioning. if you had a cheap used pump, you could bury some pipe with coolant in it and pump actual cooler less heat generated cooling into the barn. a used pump you could probably get for 2500-5k, but of course you would have to bury about 600ft of pipe about 3-6 feet deep. but, if you interconnected it with your house, you could probably run your air conditioning less.
shading devices that are not solid, ie like woven fabric, or tree leaves, let heat escape. one trick i did for the shade on the house, is not connect it about 6 inches from the house. i connected it there with only the support beams. this lets the heat escape there. most overhangs don't have a heat escape, and are therefore more effective is shorter. they can actually be oppresive if they are too larger, as the air doesn't move as well.
my ash trees i planted on the south side will propbably start to shade the house in a couple of years. at the peak of the day, the shadow is about 5 feet from the house. later in the summer, it will be a little further. the shade helps keep the lawn cooler too. these are deciduous trees, the drop leaves, so the sun can get through in the winter on the south side. another nice thing about ash trees are that they are relatively low in trash, and drop their leaves in a short period, come in a variety of shapes and leaf colors and don't creep up out of the ground like a lot of urban trees that were planted like liquid amber. those trees along with eucalyptus should be banned from planting outside their native range.