So I guess the temperature thing is moot by the end of January?
I save apple sawdust for my meat-smoking BBQ.
I'd like to turn my compost over but it would take up 1/2 my lawn and garden to get it done.
I add fertilizer and lots of water, use an iron T-post to bore some holes maybe 16".
Essentially, it's no more than a means to get rid of many cubic yards of clippings.
One year, January I recall, I needed some compost for something.
No more than 24" compact snow on the ground, maybe -10C for a couple of weeks (nice winter days.)
I thought that I could hack into the pile with a shovel or an axe and chip off a chunk.
Started to clear the snow, hit solid ice! The heat from decomposition had melted enough snow
which froze on the top, creating a 2" thick ice dome over the compost.