As some other answers point out, the most interesting part of the question is how you handle (avoiding) meeting with current time civilizations and overpopulating because of infinite iterations.
Civilizations from an ending universe would look really different from their own specie millennia ago and wouldn't be able to integrate into their society. Also, the limited resources would mean a smaller community.
As an elder civilization that has seen the end of the universe they would be likely fatalists and if found out may force their own destruction to avoid interfering in the regular timeline.
With this setup, I would consider them living in small worlds scattered in distant stars and purposely hiding by avoiding galactic groups or clusters using stellar engines to move their whole solar system. I am supposing that if they have time travel and stellar engines, they can move their whole planet/system and would be able to "steal" suns to replenish. Ideally, they should steal the suns close to a Black Hole to avoid affecting the timeline (as objects in a black hole lose their information according to current theories)
This would allow them to handle nearly infinite recursions, but at the same time, harnessing mass from black holes (or avoiding it to get sucked in) would slowly reduce the time to heat death, so in infinite iterations, they would "eventually" end up meeting their demise.
Also want to point out that it would be OK to meet with older or newer recursions that have crossed the heat death horizon as they would simply part ways knowing better not to interact.