"The cool of the grass" makes sense, but it is a sort of poetic phrasing. The sentence you presented, however, is clunky. Usually you would not say an object 'has' a 'cool' because 'cool' is not typically used as a noun in that way. You wouldn't say "the air has cool," you would say, "the air is cool." So you while you would convey your meaning, it would be in a clumsy way.