"Cool" was firstly used in jazz
Besides linguistic and etymologic analysis, cool was firstly used in jazz. Rather than a term, the word "cool" is reportedly firstly used for the sax player Lester "Pray" Young in the early 1940s. Lester Young was mainly a swing jazzman, and the word "cool" is used for pieces in order to describe how calm and relaxed was his style.
However, the word "cool" fluctuated back and forth between the meanings "'calm and dispassionate' and 'audaciously impudent'" as Ben Zimmer puts it. In 1948, a The New Yorker article about bebop jazz, described the style, more or less, "cool" in a language of their own.
Nevertheless, the same year many jazz ecoles and critics defined the new jazz style born, which is distinctively different than bebop jazz. This style was calm, contained thinner textures, softer dynamics and rather slow tempo -more likely unhurried- improvisations. The jazz critics called it the "cool" jazz. "Hot jazz is dead. Long live cool jazz!" announced The Bridgeport Telegram. Although some controversial critics made using the other of cool, we all are sure what cool jazz sounds like.