Well, of course, “cool” is entirely relative when it’s that hot. Much as I like warm weather, that’s about when I begin to lose interest.
There are several things I like when the heat is on. Really, cotton is pretty effective. It breathes well, and when it gets sweaty it offers pretty good cooling. Trouble is, then it tends to stay wet, and that can lead to chafing, or you can get chilled if the temp drops. A better base layer is a polyester-based synthetic (polyester is naturally hydrophobic and absorbs very little moisture). I wear a lot of Patagonia Capilene 1 Silkweight T-shirts. These are super-light, highly breathable, fast-drying pieces that I think are terrific in hot weather. They even offer 50 UPF sun protection. The T-shirts run $29.
A number of clothing makers also put out tropical-weight long-sleeve shirts that are great for hot weather. I like Ex Officio's shirts, such as the Air Strip Lite ($85). It’s a super-lightweight shirt made from a nylon/polyester blend (nylon holds more water than polyester but is also tougher, so there is a trade-off). It’s long-sleeve, but the sleeves roll up nicely. And a mesh "cape" panel in back helps airflow. These are really nice shirts.
Another good choice: Royal Robbins' Expedition L/S shirt ($60), made from a reinforced polyester fabric. It’s designed for hot weather, too, and would be a nice, cool shirt. Or, Mountain Hardwear's Canyon Shirt ($60), another poly/nylon blend model, generously cut, with long sleeves that roll up nicely.
Just be sure to keep some beer on ice!