I've found a reference to "ancient time" from Google.
It's mentioned in e.g. the book Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture, by Qi S. Zhang.
E.g.
The second case is when the manifold $M$ is compact and has positive sectional curvature everywhere. Let $t_0$ be a very ancient time.
What's "ancient time" and why is the name such?