a little, slightly, etc.
completely
quickly, rapidly
Dry soil cools rapidly when air temperatures fall.

gradually, slowly
Her tea was slowly cooling in front of her.

allow sth to, let sth
Allow the cake to cool thoroughly before removing it from the tin.

considerably
slightly, somewhat
He's cooled down somewhat since this morning,

be, feel, look
The forest looked cool and shady.

become, get
It will probably get cool later, so bring a coat.

keep sth
Try to keep the drinks cool.

very
a little, slightly, etc.
fairly, quite, rather
relatively
It was a relatively cool night.

completely
Wait until the cake is completely cool before cutting.

deliciously, pleasantly, refreshingly, wonderfully
The cave was refreshingly cool.

surprisingly, unseasonably, unusually
The weather that June was unseasonably cool.

appear, be, look
keep, remain, stay
She managed to stay cool during the meeting.

act, play it (informal)
He forced himself to count to ten and act cool.
For once I felt uncertain about my real feelings. I decided to play it cool.

very
pretty

about
She's completely cool about what happened.

with (AmE,informal)
I knew I needed surgery, and I was cool with it.

cool, calm and collected
He did his best to appear cool, calm and collected.

act (AmE), appear, be, sound
remain

distinctly (esp. BrE), very
rather, somewhat

about
She was distinctly cool about their plans.

towards/toward
He was cool towards me. (BrE)
I'm sorry I acted cool toward you. (AmE)

be
It's so cool you came back!

look, sound
You look really cool in those jeans.

extremely, fairly, very, etc.
so
totally
amazingly

Cool is used with these nouns:
air, basement, breeze, calculation, cheek, climate, colour, composure, confidence, confines, detachment, drink, evening, exterior, feature, gadget, gaze, green, hand, head, indifference, jazz, kid, light, look, marble, mist, morning, mum, music, reception, restaurant, sand, shade, sheet, shower, sophistication, summer, sunglasses, temperament, temperature, thing, tone, voice, weather