Well let’s take a look at the types of stars we have. I’ll start with the largest, Super Giant Stars
These big balls of death consume themselves so quickly they only last a few million years before self-destructing as supernovae. The remaining core may form a black hole if the star was massive enough.
When these guys go supernovae they spread their gases throughout the universe and these gases coalesce to form more stars, such as several like our own. They could create other types of stars such as other super giants or the gases might float in the universe forever.
Next up, a neutron star:
The core of a giant star that, like the super giant, died several million years after its birth. Neutron stars are extremely dense and emit energy as they spin.
Eventually, after a very long time, they will ‘spin down’ and stop emitting energy. This may take billions or trillions of years (we’re not exactly sure) and the oldest known neutron stars are up to ten billion years old.
Next, Red Dwarf Stars
Little red, the runt of the cosmic litter.
The most common type of stars, they emit so little light you won’t see them with the naked eye at night.
Compared to larger stars such as the Sun, they can burn a larger proportion of their hydrogen before leaving the main sequence. They have estimated lifespans far longer than the present age of the universe. The lower the star’s mass the longer its lifespan.
From wikipedia:
It is believed that the lifespan of these stars exceeds the expected 10-billion-year lifespan of our Sun by the third or fourth power of the ratio of the solar mass to their masses; thus, a 0.1 M☉ red dwarf may continue burning for 10 trillion years.
Yikes!
When you consider that there’s going to be a lot of supernovae over the next few tens of billions of years, perhaps longer, the last surviving star in the universe may not actually be born yet.
In short, yes, there will be a healthy neighborhood of stars still burning 1 trillion years from now.