Origin of idiom "full of hot air"

The OED says it means

Vaporous or boastful talk, pretentious or unsubstantial statements or claims

And that it was originally U.S., with the first citation given from Mark Twain in 1873.

Twain's 1873 use of "hot air" in the relevant sense does not involve the longer idiomatic phrase "full of hot air." J.E. Lighter, Random House Historical Dictionary of American Slang (1997) has this entry for "hot air":

hot air n. empty, exaggerated, or boastful talk; in phr. hot-air artist one given to such talk. Also (rare) hot wind. {Despite its acceptance by earlier dictionaries, the bracketed quote in its full orig. context clearly does not exemplify the slang use of this phr.} [First three cited examples:] {1873 "M. Twain" & Warner Gilded Age ch. xliv: The most airy scheme inflated in the hot air of the capital only reached in magnitude some of his lesser fancies.} 1899 Kountz Baxter's Letters 68: Bud gave us a lot of hot air about his mother's cousin. 1899 Ade Fables 58: He talked what is technically known as Hot Air.

Lighter's contention is that Twain & Warner's use of the "hot air" in The Gilded Age "does not exemplify the slang use of this phr[ase]"—so let's take a look at the quoted occurrence in fuller context. From Mark Twain & Charles Warner, The Gilded Age, volume 3 (1874):

Colonel Sellers was as unchanged as any one Philip saw whom he had known elsewhere. Washington appeared to be the native element of this man. His pretensions were equal to any he encountered there. He saw nothing in its society that equalled that of Hawkeye ["a pretty large town for interior Missouri"], he sat down to no table that could not be unfavourably contrasted with his own at home; the most airy scheme inflated in the hot air of the capital only reached in magnitude some of his lesser fancies, the by-play of his constructive imagination.

Lighter evidently reads the authors' reference to the "hot air" in Washington, D.C., as alluding to the ambient air temperature of the city on the Potomac, not to the emanations of blowhard functinaries and legislators of that city. Maybe so—but given that Twain was a sardonic critic of political behavior, I am less inclined than Lighter to interpret the "hot air" wording literally. Also potentially relevant is the fact that a translation of Jules Verne's popular 1863 novel about hot-air-balloon travel, Five Weeks in a Balloon, had been published in New York in 1869 (the technology had been used for manned flight since 1783), so the concept of inflating a balloon with hot air would have been immediately recognizable and perhaps topical to readers in 1873.

In any event, Twain & Warner did not use the phrase "full of hot air." The earliest instance I have been able to find of that expression arguably used in a figurative sense is from "Train Talk," in the [Winnsboro, South Carolina] Fairfield News and Herald (July 8, 1885), reprinted from the Chicago [Illinois] Herald:

The smoking car of an incoming train was full of passengers. It was also full of hot air.

"Hear we're goin' to have cholera this summer," remarked one passenger to his seat-mate.

"Shouldn't wonder."

"Well in that case I think it is every man's duty to clean up an' git things in readiness to fight the scourge."

"Do you mean to do that yourself?"

"Yes, I do."

"Very good. Don't lose any time about it, either. You will find a bathroom right across the street from the depot."

Here, as in the Gilded Age example, it is possible to take the term "hot air" literally, given that (1) cholera was generally taken to be a hot-weather disease and (2) body odor is likely to be more oppressive in a hot environment than in a cool one. On the other hand, the entire vignette following the words "full of hot air" consists of a essentially "vaporous or boastful talk" (the phrase used in the OED's definition of figurative "hot air," as cited by tchrist in his answer to this question) with a comic punchline, and neither speaker in the reported dialogue comments directly on the air temperature in the smoking car.

The earliest unambiguous example I could find of "full of hot air" used figuratively is fifteen years younger than the Chicago smoking car example. From "Short Bouts the Best," a segment of "C.W. Ryder's Chicago Letter," in the Saint Paul [Minnesota] Globe (April 8, 1900):

I have heard and read comments upon the "death knell of boxing as sounded in the repeal of the Horton law," but such talkers and writers are full of hot air. With New Orleans and Denver to spread the sound of the gospel, and Chicago midway, to keep the sport at the flood tide of popularity, there is no doubt, in my mind, at least, that we have only just begun to see boxing put upon the basis of legitimate sports, and when, with a few more favorable conditions, such as have been imposed by Mayor Harrison, in this city, to refine the spirit of the prize ring, boxing will attain a popularity equal to other sports of skill and endurance.

The saying could be related to Windbag: a person who talks at length but says little of any value, as in "I think he's a pompous old windbag"

windbag (n.)

late 15c., "bellows for an organ," from wind (n.1) + bag (n.). Figurative sense of "person who talks too much" is attested from 1827

(https://www.etymonline.com/word/windbag)