There's even been a war in 1992-1993, after the collapse of the USSR [1].
Sometimes I'm still surprised of my ignorance about important facts that happened during my lifetime.
Anyway, the next question for me was: why is electricity so cheap there, and should I assume it's cheap 24/7, and therefore generated with hydro?
Electricity is largely supplied by the Inguri hydroelectric power station located on the Inguri River between Abkhazia and Georgia
there's even a study on Electricty consumption, if you really want to dig deeper into it. [2].
[0]: https://en.wikipedia.org/wiki/Abkhazia
[1]: https://en.wikipedia.org/wiki/War_in_Abkhazia_(1992%E2%80%93...
[2]: http://iccn.ge/files/energy_study_results_abkhazia_2019_eng_...