What would happen if I accelerated to 0.999c seconds before the heat death of the universe? If I were to get within seconds of the heat death of the universe, and then accelerate to very close to c instantly, as I understand it, I should experience time considerably slower than a "stationary" (at least, in my reference frame) object observing me. What that leads me to conclude is that for me, it would be much longer until the heat death of the universe. Given that I accelerate to 0.99c and that for a stationary object, it would be 5 seconds until the heat death of the universe:
$$
t' = \frac{\Delta t}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}
$$
$$
t' = \frac{5}{\sqrt{1 - 0.9801}} = 35.44
$$
I should (given my amateur-ish understanding of the equation) record 35.44 seconds before the death of the universe.
However, I can see two problems with this thought experiment (really, they're most likely problems with my thinking or my understanding):

*

*The heat death of the universe has to be absolute, ie. it makes no logical sense for all the universe's energy to be completely dissipated in one reference frame 45 seconds before another reference frame

*To get to 0.99c (even if I accelerated instantly, or there about) I would have to do work, which would dissipate energy. This seems to me to mean that I would experience the heat death even faster than the stationary observer, given that a stationary observer isn't dissipating energy in any amount close to how I would if I were going at 0.99c in my spaceship.

EDIT: My example of 5 seconds before heat death is an extreme example (I imagine it wouldn't be feasible, even if we could accelerate to 0.99c) but the concept should work at any time before heat death (again, this could be entirely wrong, please correct me)
 A: Your question is based on a misunderstanding
Let’s fix that first.
Heat death of the universe
The heat death of the universe describes a future that lasts infinitely long time, during which the universe spreads out and cools further down, from its current average of about 2.73 K above absolute zero, to 1 K, 0.1 K, a thousandth of a K, a millionth of a K, a billionth of a K..... a billion billionth of a K, a billion billion billion billionth of a K, etc.
Forever cooling, forever spreading thinner and thinner, lower and lower average temperature (or average energy; it’s the same thing), less and less happening, more and more reduction of structures to simplest particles, etc.
Gradually over trillions of trillions of years, it becomes statistically almost impossible for new stars to form, because not enough hydrogen is in one place... matter and many particles break down.. more and more of space, time and matter find themselves beyond our observable universe and are lost to us forever (due to the expansion of space)...
This YouTube video, "Timelapse of the future: a journey to the end of time", is a really good illustration of just how long this takes, and how it’s expected to happen, if it does. Enjoy it!!
Your question
So when you imagine the heat death of the universe, imagine not an "event" that can be escaped, or actions taken just before it, but rather, an infinite eternal process, of continual increasing thinning, breaking down, and emptying of space and all things in it.
If you could accelerate to near the speed of light, at any time in the extreme far future of this process... you'd see a lot of ever emptier space go past, and be very bored. You'd have nowhere meaningful to go, and almost nothing would meaningfully change while you went there...
A: (1) Heat death is no switch that flips, it's a gradual process. You can't time it sharply.
(2) There is no absolute time in special relativity, so it does make sense that it takes different times for different observers to reach the same event, even if they started out synchronized. This happens, for example, with the twin paradox.
A: There are a few tiny kinks on your logic, but we can soon straighten them out! In no particular order...
You are implicitly assuming that the heat death occurs everywhere in the frame of the 'stationary' object at t=5s, where t=0 is the time when you instantly accelerate to a high speed. That's a bit of a simplification, but let's assume the assumption holds true, and that heat death occurs simultaneously everywhere in the frame of the object.
If you accelerate in a straight line from t=0, you will be time dilated in the frame of the stationary object, which means that when the local time (ie local to you) is 5s in the frame of the object, less than five seconds will have passed on your watch, as you will be time dilated in the frame of the stationary object. Given that, you might want to rethink how you reached the answer of 35.44 seconds.
The next thing to bear in mind is that there is only one underlying physical reality in special relativity, so try not to imagine that a given even occurs in one frame and then occurs at a later time in another frame, as if it has occurred twice. The event occurs when it occurs- observers in different frames don't disagree about that, they just use different coordinates to label it.
Specifically, in the frame of the object the same value of the time coordinate, namely t=5s, is used to label the location of the heat death in every point in space. In your frame, the heat death is not simultaneous, but lies on a sloping plane of time, so at every point along your direction of travel it will occur at a different time in your frame.
Another point to bear in mind is that all motion is relative. Once you have accelerated to 0.99c relative to the object, the object is moving at 0.99c relative to you, and the object will be time dilated in your frame of reference to exactly the same extent that you are time dilated in its.
Finally, in connection with your second bullet point, if hypothetically you were to hasten the onset on the heat death by accelerating, then you would hasten it in all frames. That goes back to the point about there being only one reality.
I hope that helps!
A: 
If I were to get within seconds of the heat death of the universe […]

If there is a meaningful you to to speak of (with the equipment to accelerate to 0.999 c), you cannot be seconds before the heat death of the universe, by any meaningful definition. One characteristic of the universe at heat death is that it is macroscopically homogeneous, so it certainly contains no “you” or spaceship.
Therefore it does not make sense to think about your spaceship journey accelerating the heat death.
A: Imagine you could survive somewhere between the galaxies in a station with enough energy, food and drink. You have somehow managed to become immortal.
The point is reached when all matter, except you and your station, has turned into black holes.
The holes radiate away photons, and there will come a time when only photons are left, carrying the light memories to a universe once teeming with life.
Now what will happen if you start accelerating one day before the last hole is turned to photons completely?
You reach 0.999 c 5 seconds before the last hole gives one last bright flash.
Which means, according to me, sitting there in space also. In a comparable situation. I see you reach c and 5 seconds later I see the hole give of the final breath. The hole is close to us. According to you, it takes 35 seconds, according to me just 5.
What's the problem?
