Will the conditions that allow life to evolve still be around in a trillion years? Some billions of years ago, systems like the Earth couldn't emerge and therefore life couldn't arise, because not enough of the heavy elements had formed yet. 

Assuming there'll neither be a big crunch nor a rip, stars will still be forming in a trillion years.

So the sun is one of the earlier stars in the lifetime of the universe.

It seems weird to be living at the very start of the era of the universe in which life can evolve. 
If a conscious being was chosen at random, the odds of it being on one of the first planets that allow life to evolve seems very slim! 

So I'm curious about whether some evidence indicates that life will be less likely to emerge later in the universe.

Will stars of a very late generation (the sun being a 3rd generation star)) be similar to the sun, keeping the possibility open for earth-like planets to form?
According to the 1997 paper "A Dying Universe: The Long Term Fate and Evolution of Astrophysical Objects" (arXiv version) by Fred C Adams and Gregory Laughlin, which was the basis for their book "The Five Ages of the Universe", the stelliferous era (the time when star formation is ongoing) is likely to last until ~1014 years after the Big Bang. Their estimated gas depletion timescale for the galaxy is 1012–1014 years. They do note that for the stars born after ~1012 years, metallicity effects are likely to result in significantly shorter stellar lifetimes.

So it seems likely that there will still be conditions to allow life to survive in a trillion years time, unless you happen to be using the long-scale trillion.