An "upper ceiling" for thermodynamics? Roger Penrose said in "A Road to Reality" (p.701): 

“There is a common view that the entropy increase in the second law is
  somehow just a necessary consequence of the expansion of the universe.
  This opinion seems to be based on the misunderstanding that there are
  comparatively few degrees of freedom available to the universe when it
  is ‘small’, providing some kind of low ‘ceiling’ to possible entropy
  values, and more available degrees of freedom when the universe gets
  larger, giving a higher ‘ceiling’, thereby allowing higher entropies."

He concludes thereby: 

“There are many ways to see that this viewpoint cannot be correct. It
  implies for example that, in those universe models where there is a
  collapsing phase, the entropy necessarily starts to decrease, in
  violation of the second law.”

And that: 

“This cannot be the correct explanation for the entropy increase; for
  the degrees of freedom that are available to the universe are
  described by the total phase space PUThe dynamics of general
  relativity (which includes the degree of freedom (which includes the
  universe’s size) is just as much described by the motion of our point
  x in the phase space PU as are all the other physical processes
  involved. This phase space is just ‘there’, and it does not in any
  sense ‘grow with time’, time not being part of PU. There is no such
  ‘ceiling’, because all states that are dynamically accessible to the
  universe (or family of universes) under consideration must be
  represented in PU. It may take some while for x to reach some large
  coarse-graining box from some given smaller one, but the notion of an
  ‘entropy ceiling’ is inappropriate.”

My question would be: how deeply are expansion and thermodynamic "forces", like most prominently entropy connected? Is Penrose right once again and if yes, how so? 
 A: I think Penrose is just trying to use some logic to contradict the common view. My problem with his logic is that the premise is the conclusion. If the entropy increase is caused by the expansion of the universe, then in a universe model where there is a collapsing phase, perhaps entropy will no longer increase and actually start to decrease. I’m not saying this is a valid theory, I’m just saying that our “laws” of physics are only laws because we observe them. When he says “in violation of the second law,” he is using the conclusion as the premise. The common view makes the second law the conclusion from the premise that the universe is expanding. If you change the premise to be that universe is contracting, you can’t use the prior conclusion to contradict the prior premise... it makes no sense.  
Who is to say that if the universe starts contracting that entropy doesn’t flip? Certainly not me, but it’s interesting to think about and perhaps that is what Penrose wants us to contemplate.  
I once had a physics professor explain entropy by spilling his soda on his desk. He asked us if the molecules of soda will ever be able to fully return to his cup. We all agreed that it’s impossible. He said that’s because entropy is always increasing. But what if when the universe starts contracting, his soda molecules start to return to his cup? If you think this sounds ridiculous, and that regardless of the expansion or contraction of the universe you wouldn’t expect the soda to return to the cup, then you probably agree with Penrose that there might not be a connection in that sense.  
It’s perhaps more likely that, the expansion of the universe isn’t the premise, but the conclusion. That it is one of the results of the entropy increase, rather than the cause. The universe is the soda spilling out of the Big Bang cup.
