As this answer points out, any human would first freeze rather than experience the heat death. However, assuming hypothetically that we could make some robot live at such low temperature (or even considering a theoretical observer whose perception rate depends on the speed of processes taking place inside its "body"), how the heat death would look like from its perspective?
For example, if we could say that movie characters perceive things, playing using half speed is slow for us, but from their perspective everything behaves normally. We could pause the movie and then make it play again and they wouldn't even notice.
Is the heat death in any way similar? Or perhaps no such robot is possible because its components would dissolve first, hence, no (in a way) conscious process is able to observe their own heat death?
(I was trying searching here and elsewhere, but nothing came up. It's possible that I've used wrong search terms, in which case I would appreciate someone pointing me to the correct ones.)
Edit: For clarification, I'm interested most in what is observed as heat death happens (for example, just an $\varepsilon$ before it). After the heat death happens, as the name suggest, the universe has died, so no robot can do anything.
To use the movie example, suppose that we were to put a speedometer (displays a the current playing speed) inside the movie world which could be read by the characters. Then, if we were to play a movie slower and slower until stop, after the movie is paused the characters cannot move or do anything, in particular they cannot see the reading of the speedometer (similarly the robot cannot do anything interesting after the heat death). On the other hand, just before the stop their world would not be different from their perspective, only that the speedometer would read close to zero.
Best Answer
From a thermodynamical point of view, living beings are able to reduce their entropy by exporting entropy to the external world. This does not contradict the 2nd principle, since living beings are open systems. For this reason, in a thermodynamically homogeneous universe (heat death), no change in the entropy can occur, and consequently no living beings (nor sentient beings) can survive.
But you asked about a non-living "robot", or whatever. In this general case, observing the universe means that the robot has to acquire information from the external world. It has to modify its internal state in order to store this information (for example, a picture of the universe, a measure of some physical quantity). This change of the information stored in the internal state corresponds to a decrease of the internal entropy of the robot. But again in this case, one cannot produce a gradient of entropy in a thermodynamically homogeneous universe. This would contradict the 2nd principle of thermodynamics.
One can say that the act of "observing" (which does imply a change in the information stored) the heat death is not compatible with the 2nd principle of thermodynamics.
Edit: Ok, let us suppose that the universe is thermodynamically homogeneous except for a very small region around the "robot", for a period of time $\epsilon$ before the heat death $t_0$, at which time the whole universe (including the "robot") is dead. It is a rather unprobable situation but let us assume so. At this point, what does the universe look like? All thermodynamical quantities are homogeneous, including temperature, density, and so on. Hence, there will be no recognizable structures (no stars no galaxies no anything) and all you could measure is a uniform and isotropic radiation at a temperature $T$ in all directions, similar to the cosmic microwave background. With the difference that the CMB that we measure today does reveal large-scale structures and it is not uniform in all directions. At the end of the day, the robot could describe the entire universe with a bunch of thermodynamical quantities, e.g., temperature and density. What I've written is highly speculative anyway, it is not taken for granted that the universe will undergo heat death in the first place. There are many other scenarios, I think.