We're extremely biased to think this is normal because we were born during this time period (so all bias rules apply).
Well yes, but then again it's like saying a hive of insects living near a reactor construction site is biased when it comes to nuclear energy (they might think a non-fission world is the only world). It's out of their league and they'll be long gone before it becomes an actual thing, so being biased in this sense is a very mortal thing to do.
The time where the universe is dead will be the normal state of the universe for a very long time, right now we're living at the moment where the universe is at the state of growth, birth and energy.
Time may very well end (or end up in other states of localized weirdness) before the actual projected thermal death of the universe. So there's no way of narrowing down a total time budget to make a projection of what is spent where, other than to speculate.
If we look the other way, in the femtoseconds after the Big Bang and the matter/anti-matter "war", we see a world with different rules in which we could never operate or exist. Equally strange to the endless void of the future, but where quantum fluctuations and ever so slight anisotropies rule everything, including our own existence millennia later.
But there is plenty of weirdness even today, just not at our scale, which brings me to the next point...
There are random chanches that within the void where everything vanished, random PLANETS (without any sort of sun) may appear for only femtoseconds as gasses forming themselves around random places at random chances throughout the dead universe (the connections they make are also random).
Do you want to tell them about spontaneously occurring microscopic black holes (aka virtual black holes), or should I? Who knows what eldritch horrors may spawn from the recesses of quantum space and make their presence felt when the universe will eventually undergo a "phase transition" beyond matter as we know it today. Random planets may just be the docile cows of those times.
Btw, some fun reading: https://en.wikipedia.org/wiki/Boltzmann_brain