More exactly, we don't use "the" when speaking in general terms, only when speaking in specific terms.
So "the balance of nature" --"nature" is all nature; no "the."
"The nature of the cat" -- different animals have different natures, and we're talking about the particular nature of one specific kind of animal.
It does sound like "Gleileichtgewicht," with the 'lei' in 'Gleich' repeated. If Duo hadn't used this word - in fact this exact phrase - so many times already, I might question whether I was hearing some brand-new word. It sounds like a skip on a vinyl record. Impossible in TTS, I know, but that's what it sounds like. :)