He might have attempted to swim if he wasn't 1) self conscious of taking the cloak and 2) better at swimming than a complete beginner.

For now, he'd deal with the heat. Warmed up was an understatement.

"Hnn yes, I haven't made use of any of the clothes I've been given thus far." He'd managed to find places along the way to wash his cloak, especially because of the rain.

"I am more than warmed up" You could tell he was talking about the weather.

"Let us begin"