What is the argument? Take your favorite theory that violates the Lorentz symmetry. In this theory, it is virtually inevitable that some particle species have higher speed limit than others; the speed of light is no longer universal.
If you consider a black hole of a given mass, you will find out that the Hawking temperature will depend on the particle species, too. Let me take too species for which the Hawking temperature is cooler and warmer, respectively. Let us call them the warmer species and the cooler species (or particles).
It is possible to surround a black hole by two shells constructed out of the cooler particles and the warmer particles, respectively, in such a way that the energy will flow from the cooler particulate shell to the cooler event horizon (because the shell is still a bit warmer than the cool horizon) while the energy will flow back from the warmer horizon of the black hole to the warmer particulate shell.
When you combine these two processes, it is clear that the net effect is a flow of energy from the cooler shell to the warmer shell: the black hole helped us to catalyze a classical violation of the second law of thermodynamics.
The temperatures are ordered in this way:
Warmhorizon > Warmshell > Coolshell > Coolhorizonwhich is why the energy flows in the following way:
Coolshell → Coolhorizon → Warmhorizon → WarmshellNote that only the middle arrow among the three arrows above, the flow from the cool horizon to the warm horizon, goes from a cooler object to a warmer object. But it is allowed because it is a flow of energy inside the black hole.
Such an outcome shouldn't be possible - because the second law can be easily proven informally for any Hilbert space, independently of any detailed properties of your objects and black holes - and there may exist several principles that prevent anyone from building similar gadgets in Nature. Lorentz symmetry must hold or the interactions between the shells are too strong and break the device (c.f. the weak gravity conjecture) or the shells cannot be built because of other reasons. I would personally bet that pretty much all these principles hold and must hold in Nature.
At any rate, this is another example showing that the "anything goes" approach does not apply to quantum gravity and if someone rapes some basic principles such as the Lorentz symmetry or any other law that is implied by string theory, she will likely end up not only with an uninteresting, ugly, and umotivated theory but with an inconsistent theory.
And that's the memo.