Why does the scale factor $a(t)$ in FLRW metric has only temporal dependence but not directional?
If $a(t)$ had a directional dependence, I think it would violate the cosmological principle since
The cosmological principle is usually stated formally as 'Viewed on a sufficiently large scale, the properties of the universe are the same for all observers.
So it makes sense to state that it has no directional dependence to preserve the homogeneity and isotropicity of the universe considering that FLRW metric describes homogeneous, isotropic and expanding universe. Is there a more fundamental/mathematical reason not to assume a directional dependence in scale factor $a(t)$?