No.
According to GR the distances can continue to expand indefinitely without a continuous input of energy.
Why? According to what law of nature? It is a human prejudice that things must have a beginning and an end. Because we individuals are born and then die we tend to think everything must be like that

and because we build things and they eventually fall down...
The Universe does not have to work like that.
GR is 100 years old, our theory of gravity/geometry-and-matter-interaction. For 100 years people have tried to show it is wrong. They have thought up tests. GR has passed all the empirical tests so far.
And according to GR a spatially finite universe can expand indefinitely without an input of energy.
Of course that means it gets cool and dark---little by little it gets colder and darker as distances expand. But it does not END. It just continues expanding and getting colder and darker and emptier.
If you don't like this, then
1. you have to invent a theory that passes empirical tests better than GR, agrees with present-day observations better. Many people have tried without success.
2. or you have to wait patiently and use the existing model, until somebody else develops a replacement for GR that works better and gains acceptance by the scientific community.
You cannot simply impose your own human preconceptions on nature.