I doubt this is "peat moss." However a lot of the cushion mosses need different conditions than what we can provide them in a vivarium, depending upon the species and where it came from originally. Where did you get it? Some need cooler temperatues, more light, better ventilation, etc. than what we provide in a dart frog tank.
Moss is also especially sensitive to calcium dust, so if you are feeding fruit flies dusted with vitamin supplements (which you should be doing) this may be part of the problem. Also, your frog may be simply trampling it down. Both of these problems are suggested by the fact it is still growing in the corners.
I've never had much success with the pillow-cushion mosses myself, even the tropical ones. They seem to grow where they want to grow, if at all. I've had the most luck with Java moss, but it gets rather long and needs trimming back and controlling where the frogs don't matt it down. This photo shows P. vittatus feeding among some Java moss.

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Patty
Pahsimeroi, Idaho
4 D. auratus blue
5 D. galactonotus pumpkin orange splash back
5 D. imitator
6 D. leucomelas
4 D. pumilio Bastimentos
4 D. fantasticus
4 P. terribilis
4 D. reticulatus
4 D. castaneoticus
2 D. azureus
4 P vittatus