Edit: I believe at least one of cited the authors, Dr. Carrol, no longer holds the position that this answer is based upon.
Is the eventual heat death of the universe due to the expansion of the
Firstly, I'm not a physicist or astronomer, but just enjoy reading popular science books, like those by Prof. Sean Carroll.
There's one idea that says yes, possibly, and vice-versa.
If the vacuum energy exist homogeneously throughout all of space and has a negative pressure, it would help explain why the universe's expansion is accelerating. In empty space where there is nothing but the vacuum energy, it would have a repulsive effect, and push the other empty space away from it, creating more empty space.
In the new empty space, there will still be the vacuum energy, accelerating the process.
In the system of the universe, the smaller systems like galaxies with matter and dark matter overpowering the vacuum energy, they'll remain intact, but the empty space between them and other galaxies will continue to expand.
In one way, I think this makes heat death happen faster, since eventually every galaxy will be so isolated by the expansion, that it effectively will no longer be able to reach any other galaxy once the expansion has grown to the point beyond which light coming from or going to it could travel, given its speed has limit.
Effectively this creates a new system with an observable horizon the size of a universe in its own right. But with much less energy in it. Instead of having all of the energy in the universe, it only has the energy of its galaxy and the empty space up until its observable horizon. It's radiation energy would also more easily dilute away.
As called2voyage pointed out, entropy will always increase, and one way to do this is by finding thermal equilibrium. So when the small amount of energy within the galaxy has its volume replaced with empty space, it would be much easier for it to find thermal equilibrium than if it were in a volume that had other galaxies and patches of dark matter to interact with.
In regards to why this heat death might lead to the acceleration of the universe, it turns out that even when the universe is in thermal equilibrium, it still has a temperature greater than zero due to quantum fluctuations of particles and anti-particles popping in and out of existence.
Given an unimaginable amount of time, instead of a simple particle and anti-particle fluctuating into existence, something more complex might emerge, like a pencil, or a Boltzmann Brain, or another universe.
However, these seem to be going towards a lower state of entropy, going against the second law of thermodynamics.
Instead, it would be much easier1 for an inflaton to fluctuate into existence and cause this sudden low-entropy event, to actually be a way of increasing entropy.
The inflaton has an enormously high energy density, and is rapidly expanding. As it inflates, its energy density gradually decreases. As I understand it, this decrease is not a direct relation to the increasing volume of space, but just the way the inflaton behaves.
At some point, the energy density lowers to a point where energy turns into the types of energy fields we're familiar with, and eventually particles. This is also when inflation stops and reheating, baryogengesis and recombination happen, leaving behind a cosmic microwave background; the moment when the universe cooled enough to become transparent (where as before the CMB's imprint the universe was too hot and dense for light to not bump into something).
What's interesting about this idea is that it allows entropy to always increase by taking the maximum entropy state of the universe (heat death), and further increase it with a quantum fluctuation of an inflaton particle–a much more likely fluctuation than a Boltzmann Brain or another universe–and create another big bang.
Because the entropy density of the background is so low, it is easier
to fluctuate into a small proto-inflationary patch than into a
universe that looks like ours today.1
This new universe too, will eventually meet its heat death, and find thermal equilibrium; creating the environment for another inflaton to fluctuate into another big bang, and continue increasing entropy. Given enough time it seems imply there will be an infinite number of universes as a result of an ever increasing entropy.
Which one we are in would be mystery, but what matters about this point is that it would explain why the universe started in a low entropy state; it actually started from a maximum entropy state, but fluctuated into a low entropy with a relatively simple quantum fluctuation.
We therefore believe that inflation does provide natural initial
conditions for the universe we see, once we place it in the proper
context of a larger spacetime that is stubbornly trying to increase
Due to the random nature of quantum fluctuations, this new universe will be different from ours. It will still be largely homogenous, and have a CMB that looks as smooth as ours, but it won't be duplicate of ours.
Carroll, S.M., Che, J. (2005) Does Inflation Provide Natural Initial Conditions for the Universe?