I am talking about convergent evolution that some (or all) species could evolve more accurate condititions for DNA replication and transcription. (Not into one species
)
When I mean stable, I don't mean rapidly changine. You can still have animals that don't evolve, yet cope well with seasons, it's not like their DNA spontaneously mutates when it comes to winter.
Climate variations in some areas are stable (e.g. islands where the water around stabilises the temperature.
Techtonic effects don't happen everywhere, some land masses don't even lie on any techtonic boundary. And the predator-prey relation ship maintains itself generally, if you look at graphs mapping time against number of animals, the predator and prey line fluctuate, with the predator line being slightly after.
Overall I am saying that in an environment which is not rapidly changing, species would evolve more accurate means of copying DNA. This would then propagate around the DNA pool, as there would be more copys of it and less mutation when copying. Think of this as "I it works, don't fix it", as if the animal is doing well and there is little need to change, it is guarding against bad mutations (such as sickle cell aenema and other ,especially resessive, genetic diseases) which could theaten the population.
Also remember that you do have variation in the population, and natural selection would continue evolution.
If the environment was changing, then this would be a disadvantage, having lower mutations, as there would be a lower probability that an animal will have a unique adaptation allowing it to survive and propagate that gene.