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This article cites a bunch of articles (I haven't been through them) that the effect of the two-fold cost of sex is "reduced" in stable environment or in K-selected environment. It says:

[..] the infamous ‘two-fold’ cost may be reduced or, perhaps more accurately, more easily balanced in K-selected species. The ‘cost of males’ is with respect to reproduction. Abugov (1985) argued from life history theory principles that the two-fold cost of sex might be easier to pay with K-selection. The two-fold fecundity advantage of asexual reproduction is always a two-fold fitness advantage, regardless of whether it is in a K- or r-selected environment. However, a survival advantage is worth more under K-selection than under r-selection. Consequently, it is easier to balance the two-fold cost under K-selection if sexually derived genotypes have a survival advantage.

I think that k-selection, k-selected environment and stable environment describe (more or less) the same kind of environment. Let me know if this sounds wrong to you. Here is wiki for r/K selection theory

From the same article K-selection is defined as:

K-selection might favour slower development, greater competitive ability, delayed reproduction, higher survival rates, lower resource thresholds, leading to ‘efficiency’ (rather than productivity) and constant population sizes at or near carrying capacity of the environment.

It sounds really strange to me. Can you please help me to understand why the two-fold cost of sex is lower in stable environment than in unstable environment?.

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r/K selection has become more of an heuristic for characterizing a species' strategy of reproduction and less a matter of predicting what sort of species will emerge from a given environment as it had originally been proposed by MacArthur and Wilson in 1967. The paper you cite tries to determine whether rotifers become more r or more K selective in different environments.

The thought here on k-selection is not almost from the definitions of the term, which is why it isn't explained in the paper.

In R-selected strategies, many more offspring are produced, but less investment is made in each one. Think about a cockaroach laying thousands of tiny eggs. The hatchlings might be eaten by their own mother; they are so 'cheap' the insect who layed the eggs doesn't have any anxiety about their survival. In extreme cases, most of these hatchlings will not reproduce, they may even be a bit of food for the ones which do reach maturity.

In k-selected strategies, very few offspring are produced. In addition to the 'cost of sex' limitation, they are invested in so heavily that the female may stop being fertile for a while. An example would be how female mammals which are lactating are not fertile or seasonal mating like many birds do; the females turn off reproduction so that they can focus on the success of one or two produced offspring.

k-selection is advantageous over R-selection when more of their investment pays off. If one out of ten R-offspring survive and one out of four k-selected offspring survive, then for these competing rotifers, the k-selected strategy is advantageous.

The cost of sex is that its half as efficient as asexual reproduction because males don't really reproduce - they only provide genetic material for females. You can argue r/K selection independently of this factor, but the article is suggesting that k-selection can be so advantageous as to also give sexual selection an advantage over asexual selection. I think asexual reproduction could be k-selected as well, but that seems to be the point being made here.

The reason that k/R has toned down in importance in the past couple of decades is that it has been found that individual species can adapt to shift their k/R strategems.

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