Unimodular gravity and Lovelock theorem I was reading about Unimodular gravity, this is a modified theory of gravity that  postulates that the gravity is only invariant under volume preserving diffeomorphism. So it breaks the full diffeomorphism invariance.
And there is a theorem by Lovelock that says that modified theories of gravity can be classified in four different ways , depending on the requirement of the Lovelock theorem that they violate
1) Theories that add extra fields to Einstein field Equations
2) Theories that include higher order derivatives of the metric in the action
3) Theories that add extra dimension.
4) Theories with non-locality or violation of Lorentz-invariance.
However I can't see where is the place for Unimodular gravity in this clasification, I think that Unimodular Gravity is in another classification, maybe a number 5)  Theories that break the full diffeomorphism invariance.
I'm correct?
 A: Unimodular gravity is classically equivalent to standard Einstein's general relativity with a cosmological constant (CC), the only difference is that CC in standard gravity appears directly in the action while in unimodular gravity it is an integration constant. So rather than providing a new way to modify gravity this is an example of the Lovelock's theorem at work: starting with an additional assumption we still arrive at Einstein field equations. For a proof of this equivalence see the following paper:


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*Van der Bij, J. J., Van Dam, H., & Ng, Y. J. (1982). The exchange of massless spin-two particles. Physica A, 116(1-2), 307-320, doi:10.1016/0378-4371(82)90247-3, free pdf.


In quantum theory the path integral would be different for UG and Einstein's GR, so quantum effects would allow one to distinguish between the two theories (at least in principle), see e.g. here: 


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*Alvarez, E. (2005). Can one tell Einstein's unimodular theory from Einstein's general relativity? Journal of High Energy Physics, 2005(03), 002, doi:10.1088/1126-6708/2005/03/002, arXiv:hep-th/0501146.

