I can't imagine that the new growth would survive, but I don't know what your lowest temperature in the coming months could be. Here in western NY, it could drop below zero if we get a really rotten spell. It was 60 yesterday, and in the fifties today. My peonies are still safely asleep, but my hydrangeas are loaded with fat leaf buds. The baby green leaves will surely all die, and I'll bet I can kiss all chance of flowers goodbye. They don't do well when it is a bad winter anyway.
If you only have a few peonies, could you put rose cones on them? But the lack of light may doom them in that case.
What a terrible problem! Peonies are my absolute favorite flowers, and I'd be so depressed if I mine were damaged.
My sympathy to you!
~Bunnycat

I know exactly what the lowest expected temperature might be in the coming months and I can imagine that the new growth might survive. The lowest temperature in the next couple of months might reasonably be expected to be around 10F, maybe a little lower or maybe never getting that cold. Emerging peony foliage is extremely hardy and should be fine, it is buds which appear early that are likely to be damaged to late freezes. Record lows are around 0F and that might cause some damage but it won't kill the plant.

If you have time and still have some of last falls leaves that are still crisp, on the times that you know the temperature will be really low mound the loose leaves around the plant. Do not place directly on the stems as it could cause rot. If the buds are just breaking the surface place something to provide shade to the ground. This will reduce the temperature slightly retarding growth. Could be a decorative garden piece or a piece of lattice from the lumberyard.

My buds are also showing, though I have not got any growth height. These are on my newly moved peonies, so at least I know they are alive! I raked up a bunch of leaves from a fence corner, tossed them on the buds. Our weather is also acting very odd, pretty warm for Jan. We had lost all our ground frost, but it is starting to get hard again.

I have all of the Peonies moved to new places, will be interesting to see who blooms this year. I moved them to totally new beds, new locations. I also did some tree limb trimming to get them more light. I think the new beds will provide better homes, less water logging because they are higher with better drainage. Just the one place was showing buds.