Wouldn't the heat be a huge problem? If it was converting matter at speeds approaching anything like the nightmare grey goo scenario, it would be releasing enormous amounts of heat - probably way too much for it to withstand.
Well, it depends on what gives you nightmares - it being fast, or it being relentless.
One thing that gets in the way of (terrestrial) life at high temperature is that proteins aren't very heat-tolerant. Grey Goo doesn't have to rely on fragile protein folding, so we can imagine its operation being more tolerant of high temperature.
Also, Grey Goo doesn't have a
body. Animals have to maintain bodies to contain complex structures, and that gives us issues moving heat into, or out of, those structures. Grey Goo does not use macro-structures, and we'd assume that all the chemical action would happen where the Goo touches other materials. Any other bulk is of no use to the goo - so it maximizes its surface area to maximize that interface, and spreads out thin. Really thin, like a millimeter or less, where radiative cooling is more sufficient and efficient.
Life is already very efficient at converting matter
Compared to what? And efficient in what sense? What doesn't life waste in converting matter?
"Efficiency" does not actually mean anything in and of itself - efficiency is always measured in some specific sense - like car fuel efficiency is usually listed as distance per unit of fuel used. But generally speaking, increasing efficiency in one sense sacrifices some other measure of performance. A fuel efficient car typically isn't fast, so it isn't
time efficient, for example.
In a Grey Goo scenario, we can imagine the Goo winning because it is horribly
inefficient in some sense. Say, for example, that the Goo is extremely ineffecient in terms of material waste products - the Goo is grey because it only converts a small amount of the material it consumes into more goo - most of it it leaves behind as ash. Perhaps even hot ash, dumping some of that extra heat into the physical waste.