In case you miss the post below, here is a key element I point out there. In view of your statements, I think it needs spelling out.
You state
"The H-theorem (3.6.20) holds for both directions of time and Weinberg writes about the theorem: "so we may conclude that the entropy always increases". This is very easy to check. Your misunderstanding of the H-theorem is typical of the "deepest thinkers in cosmology". "
So the key point is, if entropy always increases, in which direction of time does it increase? Weinberg's derivation has no answer. I'll explain step by step.
Choose a time coordinate t. The theorem as developed by Weinberg, according to you says
dS/dt > 0. (1)
Now choose the opposite direction of time:
t' = -t. (2)
As you admit, "The H-theorem (3.6.20) holds for both directions of time" (I show why in my post of Sep. 12, 2012 @ 17:30). Hence it holds also for t'. Therefore the Theorem as developed by Weinberg also says
dS/dt' > 0. (3)
Is (1) true or (3) true, or are both true, or is neither true?
Weinberg's derivation, like Boltzmann's says both are true. It does not pick out the preferred direction of time which underlies the 2nd law of macroscopic physics.
So in which direction of time does entropy increase? Weinberg's equation (3.6.20) does not provide the answer. It can't explain the most elementary fact about everyday physics.
So where is the misunderstanding in this elementary line of reasoning? Your sardonic comments are in tatters if you can't reply convincingly.
Maybe if you look at this carefully you'll at last understand what Wheeler, Feynman, Sciama, Davies, Zeh, Penrose, and Carroll and others were on about.