The standard picture is that any static patch of de Sitter spacetime is similar to a black hole turned inside out, and it radiates by a similar mechanism, but there is no infinity for the radiation to escape to, so there is no net evaporation. Instead there's an equilibrium with a nonzero amount of Hawking/Unruh radiation in the interior.
I'm not sure what to make of the paper you linked (by T. Markkanen). On the face of it, it makes no sense that de Sitter spacetime would evolve into Minkowski spacetime. They aren't solutions of the same field theory—or if they are, one of them is a false vacuum, and you might get catastrophic vacuum decay but not a gradual relaxation.
Markkanen says
Based on thermodynamic arguments the seminal study [Gibbons and Hawking] concluded that unlike black holes de Sitter space is stable. However quite interestingly, also by invoking thermodynamic concepts in the equally impactful work [Padmanabhan] it was argued that the de Sitter horizon in fact does evaporate.
But Padmanabhan's paper is a 100+ page survey in which the argument in question is just part of one section (10.4), so the high citation count can't be taken as evidence of support for that argument. Besides, Padmanabhan doesn't argue that the horizon evaporates; he just stops short of saying that it definitely doesn't. I don't understand why he doesn't consider it obvious that the emitted radiation will fall back into the horizon at an equal rate, since there's nowhere else for it to go.
Looking at papers that cite Markkanen, I found a paper by Moreau and Serreau, published a year later, which is not very accessible but seems to claim Markkanen is wrong. But it isn't specifically about Markkanen's paper and only cites it as part of a long list of previous work.
