Do they? While I've read that there's not a lot of difference between just chilling and thinking of some hard problem (not sure if true, though - haven't really dug to verify), thinking is thinking, so I guess that makes sense.
But when e.g. a reaction time is important, I suspect, it could be quite energy-hungry. I mean, my watch sends me that I have elevated heart rate when I'm playing any fast-paced videogames, and I guess all that blood pumping surely has some biological reason (supplying more oxygen or something) and isn't just for funsies. I'm not a biologist, so this is purely based on my own observations and stuff I've heard somewhere.