Old English col "not warm" (but usually not as severe as cold), "moderately cold, neither warm nor very cold," also, figuratively, of persons, "unperturbed, undemonstrative, not excited or heated by passions," from Proto-Germanic *koluz (source also of Middle Dutch coel, Dutch koel, Old High German chuoli, German kühl "cool," Old Norse kala "be cold"), from PIE root *gel- "cold; to freeze."
Attested in a figurative sense from early 14c. as "manifesting coldness, apathy, or dislike." Applied since 1728 to large sums of money to give emphasis to amount. Meaning "calmly audacious" is from 1825.
Slang use of cool for "fashionable" is by 1933, originally African-American vernacular; its modern use as a general term of approval is from the late 1940s, probably via bop talk and originally in reference to a style of jazz; the word is said to have been popularized in jazz circles by tenor saxophonist Lester Young (1909-1959). Cool-headed "not easily excited or confused" is from 1742.
c. 1400, "moderate state of cold, coolness," from cool (adj.). Meaning "one's self-control, composure" (the thing you either keep or lose) is from 1966.
Old English colian, "to lose warmth," also figuratively, "to lose ardor;" cognate with Old Saxon kolon, Dutch koelen, Old High German chuolan, German kühlen, all from the root of cool (adj.). Transitive meaning "to cause to lose warmth, reduce the temperature of" is from late 14c. Related: Cooled; cooling.
Figurative meaning "abate the intensity of" is from c. 1300. To cool (one's) heels" wait in attendance, "generally applied to detention at a great man's door" [Century Dictionary] is attested from 1630s; probably the notion is "to rest one's feet after walking."