Well, Rope Trick descriptions reads as follow:
The upper end is, in fact, fastened to an extradimensional space that
is outside the multiverse of extradimensional spaces (“planes”).
Well, what is an "extradimensional space" then?
A number of spells and magic items utilize extradimensional spaces,
such as rope trick, a bag of holding, a handy haversack, and a
portable hole. These spells and magic items create a tiny pocket space
that does not exist in any dimension.
Then, let's look at the planes that might have a different Time Flow:
- Astral Plane: Timeless
- Demiplanes: as they are mostly artificial, they can be created with a different Time Passage trait.
- In Pathfinder only: Limbo (Erratic Time), Plane of Time (Erratic Time), Purgatory (Timeless)
So, the only way for a Rope Trick to have a different Time Flow would be to belong to the Astral Plane, Limbo, or Plane of Time (obviously not the case), or to be a Demiplane.
However, as mentioned above, the extradimensional space created by a Rope Trick is specifically described as "outside the multiverse". And when we look at what the Planes are:
The planes of existence are different realities with interwoven
connections. Except for rare linking points, each plane is effectively
its own universe with its own natural laws. The planes break down into
a number of general types: the Material Plane, the Transitive Planes,
the Inner Planes, the Outer Planes, and the demiplanes.
So the Demiplanes belong to the Multiverse. A Rope Trick extradimensional space is outside of the Multiverse. (a short but interesting thread on the issue)
And if there were still a possibility that Rope Trick was considered as linking to a Demiplane (which would open an awful can of worms), you can just rely on the following:
Normal Time: This trait describes the way time passes on the Material
Plane. One hour on a plane with normal time equals one hour on the
Material Plane. Unless otherwise noted in a description, every plane
has the normal time trait.
So as a conclusion:
- A Rope Trick space is different from a Demiplane.
- Passage of Time is normal in a Rope Trick space, as in a Bag of Holding or Portable Hole.