Consider a general spacetime manifold $\mathcal{M}$ of a given dimension (usually $D = 4$). I call two physical constraints that should be imposed on any reasonable classical theory of physics :
Homogeneity (or universality) of physical laws
The local laws of physics should be the same everywhere on the manifold. In other words, the dynamical variables and fields should maintain the same relative relations at any point $\mathcal{P} \in \mathcal{M}$. Laws are the same everywhere and at any time. This is a physical constraint that has nothing to do with coordinates used to parametrize the manifold.
Coordinates are arbitrary labels, and the physical laws should be independant of the coordinates used to cover spacetime (covariance under coordinates reparametrization). This trivial constraint is not the same as asking that the dynamical variables should keep the same relationships everywhere and at any time.
Local isotropy
The local laws of physics should be the same for all orientations of the local space axis and for all local reference frames, i.e. for all observers located at the same point $\mathcal{P} \in \mathcal{M}$. This constraint is very well known as local Lorentz invariance, and is perfectly clear to me.
What is the proper mathematical formulation of the first meta-law (universality principle) above, in General Relativity? If it's the invariance under diffeomorphism, how is it different from the general covariance under changes of coordinates (i.e. changes of parametrization of the spacetime manifold), which is a mathematical triviality? (any equation that is true in one coordinates system is also true in all other coordinates systems and can be cast into a covariant formulation, with the use of some proper tensorial definitions).
How homogeneity (or universality) of physical laws is mathematically formulated, without any reference to a coordinate system? And how to express that constraint in a fixed coordinates system?
