The heat in a compost pile is generated by bacteria working on the materials available. If there is not a Carbon to Nitrogen ratio close to about 30:1 there will not be enough food for those bacteria to work so they stop and the compost goes cold. Volume also is necessary to some extent and most people I know with compost piles smaller than 3 x 3 x 3 often cannot maintain a hot pile for very long.
To "restart" the process, get those bacteria working faster and generating heat you need to look closely at the Carbon to Nitrogen (C:N) ratio and meet the needs of those bacteria.

Here is a link that might be useful: Composting Tutorial

The composting process has not completely stopped just because the pile is not above ambient temp. It just means it is working slowly enough that any heat generated dissipates out of the pile as fast as it is generated. It has not really stopped but is just composting slowly. Eventually even a cold pile will turn to compost. But I agree with kimmsr, add more greens.

Where does your kitchen waste go? Fruit and vegetable peelings and scraps are high in N and great for the compost. You can add them all winter when there are no grass clippings.

You didn't ask this, but if you are putting a scoop of finished compost into your new batch, you don't really need to spend $ on compost starter. All the microbes you need are already in the compost.

Well, for one thing, it's probably getting very cold in your area right now. I' guessing that if you don't have the right combination, you won't get a steaming pile!
Eh, so what? There are plenty of posts about how to compost in the winter. You're in the snowy area, I'm not! You can add to your pile and have nothing happen til spring. My time is NOW! I've got leaves and all of the garden leavings right now! The only thing I don't have is rain!
Don't stress on it! Nancy

I'm always amazed at how much water it takes to really get a full bin of dry leaves wet enough to really get the bacteria going. The combination of really dry leaves + dry fall air seems to suck the moisture right out.

For example, I built two piles on Monday. Shredded leaves, kitchen waste, trimmings from perennials that are done for the year, etc. & some urea to really get'er cookin. Wednesday my temp was up to 110. Yesterday it was down to 75 or so. I gave it some more water, and was at 120 this morning when I checked before work.

And yes... my wife makes fun of my trips 'round back with the compost thermometer. ;-)

Cincinnati - the 'compost process' can go on even when there is no heat. Some of us cold compost all the time. If my compost heats up it's a rare curiosity. But I get plenty of compost in the end.

There are other ingredients in the barrel, just not in great enought qty to be substantial part of the recipe.

While I am cooking a batch in the barrel, I pile up ingredients on the ground for my next batch. I have used this "pre-compost pile" for 4 years. The bottom of this pile is fabulous compost/top soil. It occasionally heats up, but only when rainfall keeps it moist enough. Food scraps are placed here too. It is generally mostly browns by the time I add it to my barrel. Grass clipping added to the pile are generally dried by the time they go into the composter.

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...if you are putting a scoop of finished compost into your new batch, you don't really need to spend $ on compost starter. All the microbes you need are already in the compost.

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I added the compost starter after the pile went cold, thinking it would help restart the process. That is likely faulty thinking. I though the partially cooked pile had too many greens. The grass clippings were in moist nappy clumps so I added a bag of mulched fall leaves to see if that would salvage the batch.

I don't think it's just the bacteria that cause heat in a compost pile. I believe that fungi also cause heating in compost. And my experience is that if you are able to shred leaves to a small enough size, you don't need any 'greens' at all. Oak leaves shredded in my Mackissic Merry Mac shredder heated up quite nicely and continued to stay hot for weeks. What ever nitrogen was in the leaves was enough to sustain the bacterial population. The nitrogen probably got reused over and over.

In my location, we get a lot of rain. And my compost location does not get much direct sun. So with rain , cold, shade I will never have a steaming compost pile. But last year I got some. I am thinking about getting help from worms to do the job faster than the microbes.
I covered it with a tarp the other day because everything is wet already. I have also buried some leaves in the raised bed and I pray that may the worms eat them , plse. lol

Like Flora said, it just takes longer to do cold composting. I would say good 6 to 9 months.