Any black body radiates a radiation whose spectrum (a function showing how much intensity there is at any given wavelength) only depends on the body’s temperature T. That spectrum is the so called black body spectrum, or thermal spectrum:
However, there is a difference between thermal radiation of an oven and that of a black hole. Hawking has been aware of it since 1977 when he and G. W. Gibbons published about this quantum radiation being observable from any event horizon, including cosmological horizons (that is roughly where the universe recedes with light velocity, due to its expansion, relative to an observer inside of it, so that nothing further away can ever be seen by that observer).
Thermal radiation from a warm body “exists” more independently from the observer in the following sense: If you move toward the body, the thermal radiation will become more energetic (blue shifted). You “bump against the photons” in between you and the hot body, so in this sense, that light “is actually there*” in the space between the hot stove and the observer. (*This is not denying the “non-existence” of light - in fact, this whole observer dependence issue here is a further aspect supporting that "non-existence".)
The radiation due to black holes and cosmological horizons however is due to Unruh temperature which accelerated observers are generally predicted to detect. While an un-accelerated observer detects nothing, an accelerated observer is predicted to observe being in a thermal bath that has a temperature proportional to the acceleration she experiences. Hawking radiation is this Unruh radiation that the observer sees when she is held at a constant distance over the black hole, firing her rocket motor all the time in order to not fall into the black hole. The radiation is detected because the observer is accelerated against the gravitational acceleration of the black hole.
What happens if any accelerated observer stops accelerating? The Unruh radiation disappears! What if the observer above the black hole switches her rocket motor off and stops hovering, now falling free? Same thing: the Hawking radiation vanishes! (except for certain gray body factors - always nasty details)
Consider observers hovering above a black hole and receiving Hawking radiation seemingly from the black hole, while other observers are in free fall somewhere in between the hovering observers and that black hole, and these falling observers however do not bump against the photons that the hovering observers think are “really there” in between them and the black hole. Strange?
Even weirder is the quantum radiation in an expanding universe: In de Sitter space, which is what our universe will resemble after some further expansion, quantum radiation with a thermal spectrum will be seemingly coming from all directions, but it will be the same for all observers, even if they move relative to each other (as long as they move on geodesics). The today visible cosmic background radiation is also having a thermal spectrum, namely one characteristic of a low temperature of 2.725 Kelvin, but you know whether you move inside of it or not, because it is more energetic (blue shifted) from the direction that you move into. It is somewhat more “really there”; you running against it hurts a little more than letting it run after you.
Is this an important difference? Well, if Hawking radiation is thermal, we can use exponential functions valid for thermal radiation and calculate the probability of Boltzmann brains appearing as thermal fluctuations. If this holds true, any kind of weird Boltzmann brain is possible, and since there is no difference between a Boltzmann brain and a usual brain, those weird situations that Boltzmann brains find themselves in are all possible; unlikely, but possible. Even the weirdest most horrific “terrible states” would be possible.
I suggest that we become less sure about the thermal nature and independent existence of Hawking type radiation – after all, there is no experimental verification yet! I do not mean to say that Hawking radiation does “not exist”, however, we should carefully analyze the difference between these types of radiations; thermal radiation on one hand and the more observer dependent quantum radiation whose thermal spectrum depends much on assumptions stuck into the semi-classical derivation of the Hawking/Unruh radiation.
The propagators of Hawking radiation  are interpreted as the probability that the event horizon emitted the detected fluctuation of energy E. Calculating the probability P of a Boltzmann brain with energy E via the thus derived formula does not only assume a perfect P ~ exp[-E/(kBT)] dependence, to an accuracy that maybe unwarranted (e.g. exp[-1050]). It also depends on neglecting to question the validity of certain approximations and limits like integrations to infinity during the derivation. There are more troubling issues, one being the consistency of the description in the case of Boltzmann brains.
The Boltzmann brain as a thermal fluctuation of gas atoms bumping together is little more than the time inverse of a brain exploding violently after we shot lasers on it – messy but conceivable. Such a brain is not ever seemingly coming directly from an event horizon, and in fact one may argue such impossible: With what velocity are we to expect a Boltzmann brain coming at us if its whole travel from the cosmological horizon (gazillions of light years) did only add up to a few milliseconds proper time (as experienced by the brain)**? And since we need to detect it, how can we without at detection destroying it - what kind of weird detector is that to be - can it be distinguished from something that constructs the brain with the same probability? I do not see that the case for observing Boltzmann brains being observers themselves, able to at least observe for a second, has been properly argued, nevertheless, there are articles in the peer reviewed literature that take these assumptions as if they are basically as given as experimentally observed black body radiation.
 G. W. Gibbons' and S. W. Hawking: “Cosmologicalevent horizons, thermodynamics, and particle creation.” PRD 15(10) (1977)
 W. Unruh, Phys. Rev. D 14, 870 (1976)
** This is about the stability of the brain implying some reasonable size dx against its decaying on its boundary (dt ~ dx/c). If it is supposed to be a whole freak world or laboratory, E goes up further.