Your best bet would be to check out your local fabric store. Look for some black fabric and get a "feel" for it. The leather and pleather look and feel more traditional for coats because that's what coats were made for. If you're going for the shiny trench coat look, it WILL get a bit warm on the inside. Other fabrics like canvas, silk, satin, cotton, etc, not so much.
However, there is a fool-proof method to staying cool in almost any costume. Simply sew a 6"x6" pocket on the back (between the shoulder blades) of an old T-shirt. Bring a cooler with several flat ice packs to the con, and simply wear the Tshirt with an ice pack underneath your coat.
Some Axel advice: I'd suggest making the sleeves normally, then see how the final coat fits. Then sew the sleeves smaller and smaller until yo get it just right. Axel's sleeves are very thin. This gives him a distinctive scrawny-armed silhouette.
Your own personal Jeebus.
Note to self: "Napalm sticks to babies" is not a good campaign slogan