Can an evaporating black hole emit protons with an energy beyond the GZK limit? There are protons reaching the earth with energies that exceed what their interaction with the cosmic microwave background should allow.They exceed the so-called GZK limit.
GZK Limit
Could an evaporating black hole emit protons that exceed the GZK limit?
 A: Remark: the present version of this answer ignores the fact the proton is a composite particle. Check comments for ongoing discussion.


Could an evaporating black hole emit protons that exceed the GZK limit?

Short answer
Yes.
Slightly longer answer
I see no reason for why that would not be the case, but this doesn't mean there are black holes around emitting these particles right now.
Long answer
In principle, Hawking radiation means the black hole behaves as a black body with temperature given (for a non-rotating, uncharged black hole) by $T = \frac{1}{8 \pi M}$ (units with $\hbar = G = c = k_B = 1$, while the complete formula can be found on Wikipedia). Being a black body, it emits particles of all energies, but high energy particles are heavily suppressed. Furthermore, Hawking radiation doesn't discriminate between particles and emits all sorts of particles, including protons.
The suppression factor is the usual one of Statistical Mechanics: $e^{- \frac{E}{T}}$. Hence, we expect to see protons with energy $E$ when the temperature is at the same scale. This means we need the black hole's mass to be such that $E \sim \frac{1}{8 \pi M}$ (units with $\hbar = G = c = k_B = 1$). In Planck units ($\hbar = G = c = k_B = 1$), we have
$$E \approx 8 \mathrm{J} \approx 8 \frac{E_P}{2 \cdot 10^7} = 4 \cdot 10^{-7} E_P,$$
where $E_P = \sqrt{\frac{\hbar c^5}{G}} \approx 2 \cdot 10^7 \mathrm{J}$ is the Planck energy (I used the values given by Wikipedia for the GZK limit in Joules and the Planck energy in Joules). Notice that, in Planck units, $E_P = 1$. Hence, in Planck units,
$$E = 4 \cdot 10^{-7}.$$
Therefore, the black hole mass should be
$$M \sim \frac{1}{32 \pi \cdot 10^{-7}} \approx 9.95 \cdot 10^4 \approx 10^5.$$
This is five orders of magnitude away from the Planck scale (at which point the Hawking effect calculation is likely untrustworthy), so there is no immediate reason to discard the semiclassical prediction of Hawking radiation (i.e., it might be reasonable to trust those calculations). In addition, the black hole's mass is far above the energy of a single proton. I was at first worried that high energy protons would only not be suppressed when the mass of the black hole was too small to produce them, but $10^{5} \gg 4 \cdot 10^{-7}$, so this is surely not the case.
If black holes can emit such high energy protons, why are we not seeing them?
Since the Planck mass is about $M_P \approx 2 \cdot 10^{-8} \mathrm{kg}$ (again, Wikipedia), the black hole should have a mass of around $M \approx 2 \mathrm{g}$. Usually, we expect black holes to form at about a couple of solar masses. A solar mass is roughly $M_{S} \approx 2 \cdot 10^{30} \mathrm{kg}$ according to Google. How long does it take for a solar mass black hole to get down to $2 \mathrm{g}$?
The semiclassical calculation for the mass of the black hole as a function of time is given by (Wald's QFTCS book, Eq. (7.3.5))
$$M(t) = (M_0^3 - 3 \alpha t)^{\frac{1}{3}},$$
where $\alpha$ is some constant of order unity in Planck units (it is related to the energy flux emitted by the black hole). For a rough guess, let's pick $\alpha = 1$ (Planck units). We want to solve for $t$ with $M(t) = 2 \mathrm{g} = 10^5 M_P$ and $M_0 = 2 \cdot 10^{33} \mathrm{g} = 10^{38} M_P$. Hence, we want to solve for $t$ in
$$10^5 = (10^{114} - 3 t)^{\frac{1}{3}}.$$
Hence,
$$t \approx 10^{114} t_P = 5 \cdot 10^{70} \mathrm{s} \approx \frac{5}{3.154 \cdot 10^{7}} 10^{70} \mathrm{yr} \approx 10^{63} \mathrm{yr}.$$
Since the Universe is 13 billion years old ($1.3 \cdot 10^{10} \mathrm{yr}$), this is $53$ orders of magnitude older than the age of the Universe. Wikipedia quotes a slightly larger value ($10^{67} \mathrm{yr}$), so I might have missed something in the calculation, but I guess you get the point.
I don't know much about primordial black holes, but there is some hope that lighter black holes could have formed at the early ages of the Universe and are evaporating right now and emitting these high energy jets, which would be a way of seeing Hawking radiation. However, I have no clue whether the astrophysical community expects these sorts of black holes to exist or not.
