> The fact that it is regulated to a margin of 0.3% shows quite how important it is.

This is an excellent point. It can be applied to a huge number of biological phenomena, but it almost never is.

An example where this idea actually has made it into the mainstream is body weight. I read an article that pointed out that, yes, we're fatter than we used to be, but regulating body weight by measuring energy intake and expenditure is doomed to fail. Articles making this claim are a very rich genre, but I found the particular argument fascinating:

People routinely maintain the same body weight, within say 5 pounds, for years at a time. This is a level of accuracy that we do not have the technology to achieve by using measurements of energy inflow and outflow -- we can't measure dietary energy that precisely! (And modern technology is, by most standards, very precise.)

Thus, there is a very strong suggestion, based on this incredible stability, that your body is regulating itself to a certain weight. If you adjust energy intake, or outflow, other things will adjust accordingly; those are unlikely to be primary determinants of the system's behavior.

A question from a layman: Could changing body's temperature by a few percent, for a small amount of time, be used to heal from some diseases, like cancer or diabetes?

Some diseases, yes; people do this all the time and it is called "running a fever". Fevers are not induced by the disease; they are a strategy you use to kill the disease.

Cancer or diabetes, no. Diabetes is not even a "disease" in the sense of an organism that can be alive or dead.

Thanks for the answer!