Could a civilisation use “inverse time dilation” to live longer? If an observer were to live near a black hole of sufficient mass, half an hour could elapse for them, but the outside universe could progress hundreds of years. This is due to time dilation.
We know that eventually the galaxy will stop producing stars, stars will die, the universe will undergo heat death and life will no longer be possible. A hypothetical civilisation would want to maximise the amount of time it can continue to live. So is there a way for “inverse time dilation” to occur, where a civilisation could experience the universe progressing at a much slower rate? For example, could a civilisation exist in such a way where they could live hundreds of years, but the outside universe has only progressed by half an hour?
 A: 
A hypothetical civilisation would want to maximise the amount of time it can continue to live.

This is a very hypothetical statement, given that human civilization has existed for a few tens of thousand years (homo sapiens as species emerged about 300,000 years ago.) What I am trying to say is that the assumptions about the existence of civilization on the scale of stars and galaxies lifetimes and about the needs and desires of such a civilization are very very strong assumptions.
I know that this is not what was expected as an answer for this question, but I am simply pointing out some factors that a physicist might want to consider when posing a question, so as to make it meaningful.
A: It seems unfair to me too, but you can only lose time with gravity and acceleration, never gain any. (But see Would time speed up near a large body of negative mass relative to observers in micro gravity? if you're willing to allow for negative masses. Maybe the civilization has harnessed dark energy.)
That said, the aestivation hypothesis says a civilization would want to actually skip this "hot" phase of the universe as much as possible. They could get more value out of spending the resources (galaxies) available to them later in time, when the universe is colder.
A: I'll give the straightforward answer that doesn't consider trying to bend spacetime in a clever way. This means there might be more interesting answers such as "to achieve the opposite effect, create a black hole and stay close to it" (and I'll see if I can give a try at some of these later), but for now I'll just deal with "If spacetime is given, how should the people in this civilization move around it to achieve this desired effect?"
Given two events in spacetime $p$ and $q$ (reminder: in Relativity, an "event" is a point in spacetime, so it tells you the "when and where"), the longest trajectory in spacetime between those two points is a geodesic. "Geodesic" is the relativist's way of saying free fall, so in other words, you take the longest to go from $p$ to $q$ if you do it by free falling. If instead you take a different trajectory and accelerate on part of it, you'll experience a shorter time (this is what happens in the famous twin paradox, for example).
Hence, if you want the things around you to age slower, in particular if you don't want to outlive your twin, move in free fall. Any acceleration will cost you some time.
Disclaimer: OP seems to understand this well, but I believe I should add a warning in here. Relativity doesn't allow you to live longer with respect to your own reference frame. For example, if you measure time with your wristwatch, no amount of gravity or acceleration or any relativistic effect will allow you to live two hundred years. Hence, I'm interpreting this question as "How can I avoid outliving my twin and other stuff in the Universe?". I'm also ignoring the sociological details of what an advanced civilization might or not want.
A: This is not an exact answer to your question — which is very short: Stay outside of gravity wells — but rather challenges your idea that time is a precious resource for a civilization.

The exact form the end of the universe will take is, I think, somewhat debated. True, thermodynamics imply some kind of eventual "heat death", even though it'll be a rather cold heat.
But the path to that final outcome is long. Due to the ongoing expansion of the universe it may also be that there won't be that much surrounding an observer in the end, including not that much heat. However, black holes will be around for a time that is essentially infinite; the heat death is a bit like the SLS launch. It is theoretically unavoidable but takes a while.
As long as there is matter outside of black holes, that matter can be directed into the black holes, and the resulting jets of energy can be used to sustain civilizations. Of course, there is a lot of matter, and it will last a long time. Because non-black-hole matter is a non-renewable resource the civilization will use it up over time. But they can simply re-locate, becoming cosmic nomads who burn down their rain forest, bit by bit. The cosmos as defined by the event horizon is quite large.
Eventually though, only black holes — and our civilization's habitat(s) — will be left. All non-black-hole matter within the event horizon is now under control of our civilization. Anything thrown into black holes in order to harvest radiation is lost for good. Eventually, because the black holes radiate away their kinetic energy over time, they'll fall into each other, producing bursts of gravitational waves which are potential energy sources. It will not be possible to influence their trajectories in order to accelerate that process without sacrificing precious normal matter. Perhaps now is the time when the civilization must decide to live deep in a gravity well in slow time in order to live to the next black hole collision. But, rest assured: There is enough time. Time is the one thing that's infinite for all intents and purposes because the massive black holes evaporate very slowly. To give you an idea: Currently, a stellar black hole will gain mass in spite of its ongoing evaporation because the microwave background radiation delivers much more energy than is lost by evaporation. It will need a long cosmic expansion to reach equilibrium here, all the while counteracted by the growing average size of the fusing black holes.
The death of our civilization may well be when there is only a single black hole left within the event horizon, and when they have thrown their last metric ton of matter into it, for a last candle light dinner. Then it will be time to go.
