First, that has nothing much to do with inflation. For the third time, one reason inflation succeeds is that its predictions are independent of the physics of the singularity (or whatever replaces it).
As I already said, the authors use the expression "the inflationary model" for a description of the history of the universe that includes (1) a big bang, (2) an inflationary period, (3) a period of dominant radiation, (4) a period of dominant matter, followed by (5) a period of dark energy dominance. I am sorry if the authors have led me to use a non-standard expression for all of that. I thought I already explained that. What should I call it? -- The "standard concordance model of cosmological history"? OK, then.
Second, in the standard concordance cosmological model all the laws of thermodynamics are satisfied all the way down to any instant after the singularity. Precisely what happens at the singularity is unknown, but there is absolutely no evidence or theoretical reason to posit something there which violates any laws (on the contrary).
I was asking why, at that instant when a singularity appears out of "nowhere," "nothing" and "timelessness" the laws of thermodynamics are not violated? I don't believe anyone has explained that -- if there is an explanation.
Third, there exist many speculative models which remove that singularity and are consistent with the laws of thermodynamics. The (eternal) cyclic model is not among them, but the Hartle-Hawking instanton I mentioned before probably is - in fact, it's motivated in many ways by thermodynamical reasoning.
OK, that's what I was asking about. Since I have no idea what the Hartle-Hawking instanton is, I'll have to look into it. The authors have an explanation involving the endless cosmic oscillation of branes coming in and out of contact that they claim solves the thermodynamics question. I have not read that part yet -- so, stay tuned.
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Look, I am a layman, searching for answers. I can't even come close to exploring these questions from the point of view of a physicist. I am struggling to understand this stuff. There is no need for acrimony. I have mentioned my prejudice against a theory that everything, including time, instantly came out of timeless nothingness -- whatever that means. But I have no genuine way of knowing if the "standard concordance model of cosmology" is correct or not.