Gone With the Wind, of course.
Oh, hang on.
There is another, older Georgia.
See note on methodology
It took an awful lot of pruning of the lists to get down to books set in the real Georgia, the country that was known by that name many centuries before James Oglethorpe settled the Savannah river. Even then I'm afraid the results are not terribly satisfactory. The top-ranked book is non-fiction, about the most famous modern Georgian, and specifically covers his youth in Gori and Batumi until he made the big time in Russia. But I suspect that less than half of it covers its subject's Georgian years before he moved to Russia; I'd be grateful for enlightenment on this point. For what it's worth, it is:
Young Stalin, by Simon Sebag-Montefore
For similar geographical reasons, and witha heavy heart, I must exclude the Georgian national epic poem, a glorious work of chivalry by a 12th-century writer. Most of the action takes place in India and Arabia, and apart from the fact that it's written in Georgian there's not a lot to link the events directly to the country. (I believe that there is a theory that the author wrote it while on assignment in Jerusalem, and that would make a certain amount of sense.) I commend it to you anyway; it is:
The Knight in the Panther's Skin / ვეფხისტყაოსანი , by Shota Rustaveli
Alas, the same goes for a 1975 novel by the best-known Georgian writer of the twentieth century, mainly set in Brazil relatively popular on Goodreads, unknown on LibraryThing):
First Garment / სამოსელი პირველი, by Guram Dochanashvili
The top book on Goodreads which is definitely set in Georgia is an uncomfortable account of a failed 1983 hijacking of an Aeroflot plane by seven young Georgians trying to defect to the West. The plane was stormed on the orders of Eduard Shevardnadze, then head of the Georgian Communist Party; three of the hijackers were killed in the attack and the other four were executed later. It is:
Flight from the USSR / Jeans Generation / ჯინსების თაობა, by David Turashvili
Over on LibraryThing, I want to give a shout-out to one of my favourite cookbooks, which has a huge amount of incidental colour from Georgia but can't really be said to be set there:
The Georgian Feast, by Darra Goldstein
The top book on LT by ownership, as far as I can tell, which is actually set in Georgia is the first one I ever read about the country, the account of an American journalist who was there at the dubious peak of the Shevardnadze era:
Stories I Stole, by Wendell Steavenson
I am awarding today's prize to Steavenson, as her book has a decent enough presence on both LT and GR (Turashvili is good on GR but almost invisible on LT)