Gravity warps space-time, and gravity can be thought of as entropy (Entropic gravity). So, as spacetime expands, how is space-time entropy (or the inverse of gravity) affected?
Answer
Verlinde's idea of entropic gravity is that the equations of general relativity can be produced by statistical arguments, much as used in thermodynamics. It's important to be clear that it is not an alternative theory of gravity. It is a way of explaining why the equations of general relativity have the form they do.
So the entropic gravity ideas would not change the way we describe the expanding universe. We would still explain it using Einstein's equations just as we always have done. What the entropic gravity does is to justify the use of Einstein's equations.
Verlinde suggested no concrete ideas for what might be physically happening. It's one of the strengths, or possibly weaknesses, of thermodynamics that the fine detail does not matter. Anyhow it would be premature to attach any physical interpretation to his ideas. In any case you should note that his ideas are hotly contested and far from being accepted as mainstream physics.
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