This is an idea
for a method
, assuming that there is enough mass
in the universe to eventually make it collapse
under its own gravity
The death of the universe this way, commonly termed "The Big Crunch" sees the universe contract smaller and smaller, with temperatures and latent energy rising and rising until electrons are stripped from atoms, atoms fly apart into quarks and the universe becomes an opaque soup of fundamental particles. Eventually the universe would supposedly reach a singularity, where all the matter in the universe is compressed to a point infinitely small, where density is infinite. This is a total bitch for any civilisation still around and wanting to keep living.
So what is the idea for escaping this fate?
One of the recurring themes in theoretical physics, is that of unification in higher dimensions. That is it becomes easier to express more than one of the 4 forces (electromagnetic force, strong force, weak force, gravitational force) through a single method/equation if you express it in a space of higher than 4 dimensions(updown, leftright, forwardback, space-time). According to theories like supergravity, superstrings, M-Theory etc, the universe started as most likely a 10 dimensional space. As the universe expanded, latent energy decreased and these extra dimensions became unstable, curling up to sizes that make them unobservable today (i.e. smaller than the Planck length).
In the same way, when/if the universe contracts, latent energies will increase and we might see that these extra 6 dimensions (our observable universe is 4 dimensional, space-time being the 4th dim.) will uncurl and expand, and there may lie the path to salvation.
Presumably, some civilisations still around by then will be Type 3 Civilisations or greater, and thus possess enough energy to utilise the extra dimensions of "hyperspace" and create a bridge between our 4th dimensional universe and the new 6th dimensional one, expanding as ours collapses.
So the trick is to pack your entire population and technology (or indeed create the means to use your planet / floating space-ship home) as an inter dimensional vehicle itself. Then as the universe begins the final crunch and shit is getting a little uncomfortable, hit that hyperspace button and create a bridge to a new universe, spawning from the death of our own, and bug the hell out.
A sixth dimensional home would be an interesting thing to see, imagine how a 3/4 dim. universe looks to the 2d person who lives on a page. If you point your finger at him and move it in and out, all he sees is this big pink circle expanding and contracting, he thinks "how is that possible?".
The analogy of this whole hyperspace thing is to imagine us living inside a square outlined in black on a page. As the square shrinks, we may use the extra dimensions "up/down", previously unimaginable, to jump out of the square and land back down on a square that is expanding somewhere near (heh maybe this time it will be a circle).
It's just a cool thing to think about, and I recommend Michio Kaku's book "Hyperspace". I'll put the ISBN up here when I can, but you should be able to find it by searching.