> In order to have a higher temperature than the CMB a black hole would have to be really small with a mass about half of that of the moon.

Either that, or you just wait a couple eternities for the CMB to cool down enough.

Or you make a smaller one and watch it shrink. It is theoretically possible to construct a smaller black hole by cramming the necessary mass/energy into a small enough space. (Plug a death star into the LHC's big brother.) Such a hole would be very hot and short-lived.

Or primordial black holes with such a small mass actually do exist, who knows...

how can the temperature of a black hole make sense? Is it the temperature of the singularity point? Is it the temperature of the space inside the event horizon (but it can't be, as that space is empty)?

"Temperature" in this case refers to the amount of energy they emit due to hawking radiation.

By comparing to a black-body curve, you can define a temperature for the hole. Obviously it's not a real object with a real temperature -- if it were, I believe the temperature might be infinite -- but this still works for the purposes of deciding whether it'll grow or shrink.

Temperature has several definitions that produce the same number under normal circumstances but may or may not be applicable in extreme circumstances. In this case, I imagine they could be using the thermodynamic definition (dE/dS, the marginal change in energy per marginal change in entropy - I’m not sure if a BH has well-defined entropy) or something to do with the emission curve of space around the black hole. I vaguely recall something about empty space behaving like a blackbody under a gravitational gradient, so maybe they can use that. You could also use amount of Hawking radiation per surface area. It’s possible that some of those produce the same number.

My understanding is it's the temperature of the event horizon: because it doesn't emit any blackbody radiation, other than the tiny amount from Hawking radiation, any heat measurement from it would be approximately 0k.

The Hawking radiation is exactly its black body radiation.

it is a phenomenon at the event horizon at with radiation is emitted in a way comparable to the radiation produced by every object according to temperature

It _will_ happen, but first dark energy needs to be strong enough to expand the universe fast enough (faster than light) so that CMB wouldn't be able to reach anything.

Well there are already portions of space that are expanding faster than the speed of light relative to our position. (see cosmic horizon). The CMB is not just a glowing heat somewhere far away, it's everywhere in the universe in every volume of space. The moment when every point in space (on a Planck-lenght-scale i guess) will be expanding faster than the speed of light relative to one another, than space-time itself will rip apart and that's the end of our universe - at least that is what the Big-Rip theory proposes.

That's my point exactly. Maybe you misunderstood.

Finite speed of expansion is enough.

The temperature of the CMB just needs to drop below the temperature of the black hole.

From the outside they must look cold since they can't radiate heat any more than light. That doesnt imply anything about the inside.

What's "inside" of a black hole, meaning behind the event horizon is forever causally cut off from our universe. The Hawking radiation comes from the space around the event horizon. In theory black holes can become very hot if their mass is small. This happens at the end of their life which is in the order of 10^80 years for stellar black holes.

Here is a calculator to play with some values. https://www.vttoth.com/CMS/physics-notes/311-hawking-radiati...

Note that this evaporation time assumes a universe at absolute zero. That 10^80 years can't even begin until the CMBR cools enough, perhaps 10^40 years.

Admittedly that's an eyeblink compared to the evaporation time scale, but it does mean that we won't observe any evaporation until many orders of magnitude longer than the universe has existed.

Or unless you manufacture or discover a low–mass black hole

That's right. If we could create one in a supercollider, it would would be so small that it would be hot enough to evaporate instantly.

There might also be a range of primordial black holes formed directly out of pre-CMBR energy. They'd have to be small enough to be hotter than the CMBR, but not so hot that they'd already have evaporated in the last 14 billion years. That's a relatively narrow range, all things considered, but if primordial black holes exist at all then they could exist at any range.

The inside is empty space.

The inside is empty time, space waved you a goodbye at the event horizon.