You are correct, Grant, that drywall will not work. This will deteriorate quickly. Since gazebos are post and beam construction, most of them just leave the roof structure open. That way any rain leaks can just come through and you will have ventilation of the area. If you really have to put something up there, about the only thing that I can think of is T1-11 plywood. You will need to cut and install some vents in the ceiling with screening to keep insects and critters out.
The potential for rot will be much greater with the plywood up there and trying to remove wasp nests or bee hives will not be fun at all. It will hold in the heat that you want to avoid. All these factors point toward not putting a ceiling up.