Is there any example of a physical theory which isn't invariant under translations? Isn't it trivial that all physical theories in spacetime are invariant under local translations?  Is there an example of a theory which isn't invariant under translations?
Please, take note that I'm not talking about homogeneous metrics, i.e. isometry under translations.  I'm considering general theories of some physical laws defined on spacetime.  Physical laws should stay the same everywhere in spacetime, or else they cannot be called "laws".
If invariance under local translations is a triviality, can it be linked to invariance under diffeomorphisms, which can be interpreted as a reformulation of general covariance?
 A: First, 
in response to comments by OP I would like to emphasize that 'theory' is not necessarily an all-encompassing 'theory of everything', generally a theory would have external objects not described by it.
With this in mind, an answer to the title question would be 'sure'. Just start with a theory that is invariant under translations and break this symmetry.
This could be done by introducing objects (fields, sources, etc.) external to the theory that break such invariance.
For example, we could start with the Standard Model and introduce time dependence of its parameters (so that they would now vary on a cosmological timescales). This new theory no longer has time translation invariance, but is potentially testable as we could look for geological evidence or astronomical observations of these variations. Of course, we then could attempt to 'restore' invariance with respect to time translations by building a new, more universal theory and attributing this variations to a new dynamical field, describing its dynamics in a time invariant fashion. A review of the type of theories could be found here:


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*Uzan, J. P. (2003). The fundamental constants and their variation: observational and theoretical status. Reviews of modern physics, 75(2), 403, doi, arXiv.


Another example of breaking translational symmetries is an orbifold construction in string theory.  By considering quotient of a manifold by a finite group we break manifold's symmetries, and thus an orbifold would have its own string theory, in particular because there are now new degrees of freedom associated with orbifold's singular points.
A: Here are some possibilities in addition to the ones in AVS's answer: --
Physical theories don't even have to have spatial degrees of freedom, e.g., a spin glass.
You could also have a theory in which there are spatial degrees of freedom, but they're discrete. E.g., I think spin foams might qualify.
I suppose it's debatable or a matter of semantics whether general relativity has translational invariance. There certainly isn't any such thing as a translation operator (infinitesimal or finite) that can be applied to a solution of the field equations to give a new solution.
