This is a pretty simple suggestion but I think it would be a potentially very interesting idea. Basically, the premise would be that the player is able to choose when life develops on their planet relative to the age of their universe during the planet generation step of creating a new save. The age of the universe could dictate different aspects once in space stage, having different consequences for gameplay depending on when you exist.
Existing in the beginning of the universe could be more chaotic, with quasi-stars being scattered around along with still-forming solar systems, more abundant gas clouds, and a distinct lack of life. The more you play this era, the more life would begin to appear that’s separate to yours. And, because of you popping up so early in history, you could wait. Playing enough on the same save would cause the universe to age into the next phase, and you would get special status among other races for being among the first in the universe.
The next era you could exist in would be the early middle of the universe. (The era we exist in now.)
Life would be somewhat abundant but still hard to find at times. Supernovas would be rare but existing occurrences, and most systems would be fully formed and stable. You wouldn’t get any special status developing in this era since you’d exist in the most populated time of the universe.
The second-to-final stage would be the late middle of the universe. At this stage, you would exist in a universe during the beginning of its end. Supermassive galaxy clusters would be abundant, but there wouldn’t be much outside of them. Black holes would be much more common, and most alien races would be more powerful than earlier stages on average. Developing in this stage would give other alien races a sense of hierarchy over you due to the age of your civilization, causing them to dismiss your race more often.
The final stage would be the late end of the universe. In this stage, stars would be rare and black holes / dwarf stars would be much more common. Once again, the universe would be mostly devoid of life due to other races dying out long before you developed, and any races that did exist would be much more powerful than any other era’s races. The expansion and heat death of the universe would be a looming threat in this stage, and you’d have to create an ascension gate before the universe ended in order to win the game. This era of the universe would act as a challenge mode of sorts.
Finally, if the theory allows it, bringing eras of the universe into the game like this could introduce the possibility of time travel as a solution to prevent your species from having to move forward into different eras. This is only speculation though since there probably isn’t any solid theory for allowing time travel to work.