What is meant by "cools down quicker"?

Will near-boiling water drop 10 temperature points in a shorter time than the warm water? Yes.

Will it reach 10C faster than the warm water? No.

No?

Today's your lucky day, you get to learn about the Mpemba effect.

(Although the why of the effect is disputed, the trivial counter to your point is that boiling water loses mass quickly so there's less mass to cool)