You just turned your jeep into a oven. I've got a TJ that I bought with no carpet and it is miserable to drive in the summer. I bedlined it to try to cut down on the heat and it was still miserable. Bought a bedrug and still terrible.
I had it at the lease this weekend and really couldn't tell the difference in temperature with or without carpet. Could've been cause it's plain miserable either way when it's 100 degrees. It was differently easier to wash out.