The theory you have laid out here is not one I have encountered before. While it is an interesting thought, I do not think it will do. The universe certainly had a beginning and will certainly grow cold, dark and uninhabitable. I have not come across any theory in which the universe will become empty. Nor do I know of any physical law which would suggest such an ending.
We talk about an "expanding universe" because we can see objects moving away from us at an increasing rate in all directions. That's because space itself is expanding.
We assume that the "edge" of the big bang back ground radiation is the "edge" of space but we can only judge space if there is an object to measure.
"space/time" becomes meaningless when there is nothing detectable within space time.
The ever growing expansion that we are witnessing will ultimately separate all matter. It will take a VERY long time to do so, but it will happen.
Once everything has been broken down to atoms then to subatomic particles, then to whatever is smaller still, the Universe will be "empty". Nothing will be detectable.
At that point space/time ceases to be a concept with any value.
A trillion trillion trillion years could pass and nothing would occur.
So, if, as some suggest, the big bang is the result of colliding membranes and these collisions are exceedingly rare, it's just a matter of time within the endless void of no change before a new Big Bang is triggered and a new "universe" is created.
Technically, that new Universe is within the existing universe, but it would be impossible for the life forms within that new Universe to detect or measure anything happening in the "old Universe" into which they are expanding.
We _could_ be the first Universe.
We _could_ be the ten millionth Universe.
We'd have no way of knowing.