hot air (n.)

c. 1300, "invisible gases that surround the earth," from Old French air "atmosphere, breeze, weather" (12c.), from Latin aer "air, lower atmosphere, sky," from Greek aēr (genitive aeros) "mist, haze, clouds," later "atmosphere" (perhaps related to aenai "to blow, breathe"), which is of unknown origin. It is possibly from a PIE *awer- and thus related to aeirein "to raise" and arteria "windpipe, artery" (see aorta) on notion of "lifting, suspended, that which rises," but this has phonetic difficulties.

In Homer mostly "thick air, mist;" later "air" as one of the four elements. Words for "air" in Indo-European languages tend to be associated with wind, brightness, sky. In English, air replaced native lyft, luft (see loft (n.)). In old chemistry, air (with a qualifying adjective) was used of any gas.

To be in the air "in general awareness" is from 1875; up in the air "uncertain, doubtful" is from 1752. To build castles in the air "entertain visionary schemes that have no practical foundation" is from 1590s (also towers in the air; in 17c. English had airmonger "one preoccupied with visionary projects"). Broadcasting sense (as in on the air, airplay) is by 1927. To give (someone) the air "dismiss" is from 1900. Air pollution is attested by 1870. Air guitar is by 1983. Air traffic controller is from 1956.

Old English hat "hot, flaming, opposite of cold," used of the sun or air, of fire, of objects made hot; also "fervent, fierce, intense, excited," from Proto-Germanic *haita- (source also of Old Saxon and Old Frisian het, Old Norse heitr, Middle Dutch and Dutch heet, German heiß "hot," Gothic heito "heat of a fever"), of uncertain origin, perhaps related to Lithuanian kaisti "to grow hot;" both could be from a substratum word.

With a long vowel in Middle English (rhyming with boat, wrote) which shortened in modern English, perhaps from influence of comparative hotter. As an adverb, Old English hote.

Hot as "full of sexual desire, lustful" is from c. 1500; the sense of "inciting desire" is 18c. Taste sense of "pungent, acrid, biting" is from 1540s. Sense of "exciting, remarkable, very good" is 1895; that of "stolen" is first recorded 1925 (originally with overtones of "easily identified and difficult to dispose of"); that of "radioactive" is from 1942. Of jazz music or combos from 1924.

Hot flashes in the menopausal sense attested from 1887. Hot stuff for anything good or excellent is by 1889, American English. Hot seat is from 1933. Hot potato in figurative sense is from 1846 (from being baked in the fire coals and pulled out hot). Hot cake is from 1680s; to sell like hot cakes is from 1839.

The hot and cold in hide-and-seek or guessing games (19c.) are from hunting (1640s), with notion of tracking a scent. Hot and bothered is by 1921. Hot under the collar in the figurative sense is from 1895.