Two drawbacks to keep in mind, and one "neutral" consideration:
1) Egregiously expensive for the benefits you get. You have to WAY SERIOUSLY beef up the load-bearing components of walls and roof, and mess round with impermeable membranes and stuff.
2) Reasonable chance of developing semi-intractable slow roof leaks as time passes (from what I have heard from others who've done it on small structures like that)
3) You really do not get much benefit for what you're laying out $$-wise... in a LARGE AIR-CONDITIONED building it sometimes makes economic sense (if you are using a good design well-engineered for your situation and correctly installed), but for a small coop it is extremely unlikely to do very much good, because indoor coop temperature is just a reflection of outdoor temperature. (Unless you are one of the about 2 people on this forum who [claim to] have a/c installed in the coop). And green roofs are really mostly just helpful in the summertime -- in a few climates they are maybe marginally-useful in the winter but more typically analyses show that they are either neutral or make the bldg *cooler* in wintertime. (Earth is really quite a poor insulator.)
If you want to do it as a hobby thing, or to feel all eco-fashion-forward or that kind of thing, then I see no reason NOT to do a green roof (bearing in mind that unless you water it regularly or use climate-appropriate succulents, it will be more like a 'brown roof', seriously).
But if you want it as a PRACTICAL measure, you would be infinitely better off just investing in good insulation and lots of summer shade and perhaps an intelligent passive-solar-and-thermal-mass element to the design for wintertime.
JMHO, good luck, have fun,
Pat