In the early 1990's, there is a bit of excitement when there was the possibility that you had globular clusters that were older than the universe, but it turns out that that disappeared when people came up with better stellar models.
There are a ton of things that "you just don't see" then you would expect if the universe were older than we think it is.....
1) red dwarfs that are off the main sequence
2) stars with very low amounts of hydrogen
3) very cool white dwarfs
In each of those cases, you end up with a good explanation of why you don't see these things by assuming that the universe is 13 billion years ago.