Yes and No
The thermophilic heap has processed what it can with what it had to work with, so it's done being a hot heap. The mesophiles will take over from here, slowly continuing the decomposition process until they run out of resources, at which point the
fungi will take over to break down what is left into stable humus. Whatever conditions are present will determine what decomposing lifeforms will be dominant. Right now, I'd estimate the heap will support a strong earthworm population.
Adding more green material and checking the moisture will get it heating up in a couple of days. What you would be doing is offering materials which will support the thermophilic bacteria. Giving it a toss can mix the material, add oxygen and redistribute the stuff already in the heap which can also cause the pile to resume heating.
Compost can be used at any stage. You can add this to your
garden beds and it will offer plenty of nutrients, but if there is an abundance of nitrogen in the soil, heat can be generated which can kill off tender seedlings. You may want to wait until the material is no longer identifiable before adding it to growing areas. As a covering
mulch, this stuff will probably do well. The fact that the heap got good and hot then cooled is sometimes offered as a novel
composting method or 'trick' to make compost fast. In reality it is just the end of the first stage of the process. It takes several months to produce finished compost which has gone through all decomposition stages. The end product will be dark brown to black, have a crumbly texture, and earthy smell, and you won't be able to find anything like the material you put in at the start.
It sounds to me like this is your first attempt at composting. You got it good and hot for a month, so you are surely on the right track. Keep the heap going to see the entire process. In the meantime, start another heap. You can add a sprinkle of this heap to help get your next heap going-this is called inoculation. Plenty of folks use a bin system with multiple heaps in different stages so there is a continuous supply of compost.
Keep up the good work. You are doing fine.