One big downside of strawbale and cordwood (you didn't mention that technique, but I will) is that the walls are thick, so the structure ends up being either much bigger than you would expect, to make the inside what it needs to be, or much smaller on the inside because you kept the footprint reasonable.
As for cob, depends on how you're going to make it if you have to worry about wall thickness, but like Alan said, you need to allow for ventilation.
A good way to ventilate in hot climates is to have plants on the north side of the building (for evaporative cooling effect) and have a ground-level vent door open to that area. Then, have an opening at the top of the coop (this works great with shed roofs which rise to the south) on the south side, so the air convects up out.
I am not sure that in a small structure like a coop that you will get enough thermal mass cooled during the night to keep the temps down during the day, even if you did have a concrete floor. If you were trying for a lot of thermal mass, you should of course insulate the outside wall, and then bring TM to the inside. One good method of adding thermal mass without a lot of thickness is using 3 layers of drywall. That only works where it's going to stay dry, though.
One other natural material construction method (not too natural, but relatively easy to accomplish, and very sturdy) is ferrocement. I've built water containment tanks out of this method; no reason it couldn't be used for chicken containment!