  We study the consequences of the modification of the phase space structure of
General Relativity imposed by breaking the full diffeomorphism invariance but
retaining the time foliation preserving diffeomorphisms. We examine the
different 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, we
also show that these sectors allow for solutions, among them some well known
families of black hole and cosmologies which fulfill all the constraints. We
raise some physical concerns on the consequences of an absolute Galilean time,
on the thermodynamical pathologies of such models and on their unusual vacuum
structure.
