Composting is a relay. Mesophilic organisms, which live at everyday temperatures, start on the easy sugars as soon as straw is wet and nitrogen is available. Their own heat then pushes them out and hands the heap to thermophilic bacteria, fungi and actinomycetes, which prefer high temperatures and break down tougher material.
That is why a heap which will not heat is a feeding problem, not a heating problem. DMR notes that when nitrogen is kept below 1.5 per cent the heap may not pass 55 to 60 degrees Celsius, because there is too little microbial activity to make the heat.