Is it really a constant or can there still be a (a little) variation in the number when measured over astronomical distances, or over large timescales?
However, most experimental data so far is in agreement on it actually being a constant.
Wikipedia has a section on it: https://en.wikipedia.org/wiki/Fine-structure_constant#Is_the...?
One of the things the cosmologists are always doing is looking for anomalies that might indicate that some "constant" really has changed over time or distance.
Apparently there can be differences based on if you measure in Paris or Berkeley? Though it’s more likely there is some other source of error. I think having the technical ability for a precise measurement may one day help us look for fluctuations.
The flip side of that question is the larger and generally accepted as obvious statement: a constant with value "x" determines or reflects how the universe is as of now. The duration of that "now" is likely much longer than the earths entire existence.