[alboaie]

"A cold, dark universe is billions, if not trillions, of years in the future. Between now and then, humans will face plenty of other calamities: wars and pestilences, ice ages, asteroid impacts, and the eventual consumption of Earth—in about 5 billion years—as our sun expands into a red giant star. To last until the very end of the universe, an advanced civilization will have to master interstellar travel, spreading far and wide throughout the galaxy and learning to cope with a slowing, cooling, darkening cosmos. Their greatest challenge will be figuring out how to not be here when the universe dies, essentially finding a way to undertake the ultimate journey of fleeing this universe for another.

Such a plan may sound absurd. But there is nothing in physics that forbids such a venture. Einstein’s theory of general relativity allows for the existence of wormholes, sometimes called Einstein-Rosen bridges, that connect parallel universes. Among theoretical and experimental physicists, parallel universes are not science fiction. The notion of the multiverse—that our universe coexists with an infinite number of other universes—has gained ground among working scientists." From article... The infinite concept is probably a human made conncept so and anyway 1 million years is probably more than enough to change your "identity" a few hundreds time (at least). So this surving the end of the universe is a bit too much to consider even in the wildest fantasies ;)

[pfdietz]

Relevant SF story:

http://www.isfdb.org/cgi-bin/title.cgi?44207

[perl4ever]

There was also this:

http://www.isfdb.org/cgi-bin/title.cgi?1044

It was assumed that the universe would eventually stop contracting and end up in a "big crunch".

The plot is basically "bussard ramjet spacecraft gets its engines stuck and goes faster and faster until the universe ends and a new one expands".

Ignoring for narrative purposes there not being anywhere "outside" of a contracting universe.