whatsnext hit it - heat of sugars and starches being broken down are amplifed & microbial activity increased from organic chemistries occuring faster at higher temps. As temperatures pass certain points different families of microbes bloom, each wave filling its niche in the compost cycle. If temperature rise stalls in the 'thermal mass' it can block any further breakdown, one niche organism overpopulates and poisons the compost media so strongly the next 'niche' organism is suppresed.
Here in Minnesota we have 10 weeks in summer where you can make black dirt compost - sometimes taking 7 to 10 days start-to-finish to digest even the cellulose in leaves and grass clippings. Otherwise cool ground and air temps stall out compost cycle, even if you water with sugar water and add nitrogen fertilizers, leaving stuff a "dry rotted" fluff full of mychorizia (?) fungal stuff.
If you kept the mass high enough and the media well stirred, w/ constant new waste being added and the oldest media being removed, it could boost water temps in a tempering tank (etc.) but it would be very labor intensive. Automated processes required as people don't much like to face that stuff intimately on a daily basis..
I think the wood-by-product co-generation plants in scandanavia get some heat out of fermenting the huge bulk of wood chips used, if only to help dry them out some before being fed into steam boilers...