Nope. Usually I let the cookies sit on the stone for 5-10 minutes though, so it does start to cool off a little. FYI, one way to make baking batches of cookies go faster is use parchment paper. You can cut out a piece that will fit your stone and while the stone is in the oven with cookies on them, scoop out more dough onto the paper. Then once you remove the baked cookies, just slide the parchment paper onto the stone with the cookies on them. It saves time!

 

I've only done it a few times this way, but I don't remember it being any different than usual when I've used parchment paper. Maybe experiment and see what happens.

 

I don't cool my cookies at all, I place them directly on the stone, take out the first batch and start placing them (very carefully so they don't break in transfer) on the stackable cooling rack and insert the 2nd batch