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We study the consequences of the modification of the phase space structure ofGeneral Relativity imposed by breaking the full diffeomorphism invariance butretaining the time foliation preserving diffeomorphisms. We examine thedifferent sectors in phase space that satisfy the new structure of constraints.For some sectors we find an infinite tower of constraints. In spite of that, wealso show that these sectors allow for solutions, among them some well knownfamilies of black hole and cosmologies which fulfill all the constraints. Weraise some physical concerns on the consequences of an absolute Galilean time,on the thermodynamical pathologies of such models and on their unusual vacuumstructure.

Remarks on the consistency of minimal deviations from General Relativity