Correct spelling, explanation: the word coolly comes from the adjective cool where the simple rule has been applied. The suffix ly is added to the adjective without any changes. The word has its roots in German and it was used in Old English and then in Middle English as col. As we can notice, it is a very easy form to memorize. Coolly is the correct form of adverb, not cooly.

Definition of coolly:
adverb, in an unemotional and unenthusiastic way; lacking in warmth,
She greeted him coolly because he had lost her trust.
Mark was shocked but Emma took it coolly and calmly.

Incorrect spelling, explanation: cooly may be a misspelling or a typo where there is only one consonant l. But this word can be used also as a noun in an offensive way, referring to an unskilled worker. However, as an adverb, it is spelled incorrectly because it is a derivation from the adjective cool to which is added the suffix ly. As a result, it should be written coolly, not cooly.