You are confusing the cosmological principle with the uniformitarian principle [1]. The cosmological principle concerns the large-scale distribution of matter (not of laws) in the universe, and that is something we arrived at empirically, through observations with large telescopes (galaxies are approximately equally distributed in all directions) and the measurement of the cosmic microwave background. It doesn't get more empirical than that. It's not something like an axiom at all, just an approximate generalization.
> We haven't done physics experiments across the universe, so we can't call it a law because there is not enough experimental evidence.
That's equally true for the second law of thermodynamics: We haven't done physics experiments in the distant past or in the future direction, so we strictly speaking can't be certain that entropy doesn't decrease in the future. But no law is perfectly confirmed anyway (not tested in every conceivable circumstance), so that can't be a criterion for lawhood anyway.
But I already proposed a better criterion: A (fundamental) law says what is possible and impossible, and neither the cosmological principle nor the second law ("law") of thermodynamics do that.
As far as I am aware Uniformitarianism is basically the same thing but covers a bit more since it includes "natural processes" like the results of erosion and is used by Earth science types. Cosmological principle is used at the cosmological scale so they don't bother including planetary natural processes in their axioms.
> We haven't done physics experiments across the universe, so we can't call it a law because there is not enough experimental evidence.
That's equally true for the second law of thermodynamics: We haven't done physics experiments in the distant past or in the future direction, so we strictly speaking can't be certain that entropy doesn't decrease in the future. But no law is perfectly confirmed anyway (not tested in every conceivable circumstance), so that can't be a criterion for lawhood anyway.
But I already proposed a better criterion: A (fundamental) law says what is possible and impossible, and neither the cosmological principle nor the second law ("law") of thermodynamics do that.
[1] https://en.wikipedia.org/wiki/Uniformitarianism