"Instabilities" in Quantum Gravity As far as I understand, there is no notion of conservation of energy in GR: the Landau-Lifshitz Pseudotensor is not conserved in all possible coordinate systems, and thus such a notion of energy conservation is unphysical.
Hypothetically, does that mean that there may exist some mechanism in nature which could continuously create particles e.g. a time-dependent gravitational background due to an oscillating source produces more and more particles simply "creating" more and more energy (I am talking about energy arising only from matter fields, not gravitational field energy, as per my last paragraph where we abandon the notion of gravitational energy) without limit? The existence of such a mechanism would be preposterous... one could then exploit gravity to create more energy than they had in the first place.
Could someone explain to me where I went wrong? For example, does there exist a gravitational "back-reaction" which acts to suppress such a mechanism?
 A: 
there is no notion of conservation of energy in GR

There is no notion of local conservation of energy in GR, but this simply means that we cannot assign gravitational energy as belonging to a specific spacetime points and does not mean that there is no conservation laws for the gravitational energy at all.
Instead one can define (in a coordinate independent way) so-called quasi-local energy, representing the energy of both gravitational field and matter contained inside specific region. For example, if such region contains an isolated system with  long-range field falling off in the usual way associated with finite mass, then  we can define asymptotic quantities, ADM energy & momentum (calculated as a surface integrals over the asymptotic region) that would be conserved, no matter what occurs inside this system.
Quantum theory could enable production of matter fields from gravitational, for example collision of two gravitons could produce two photons or e.g. electron–positron pair, we thus have creation of “matter from space”. But such collisions must still satisfy energy conservation.
Another class of “preposterous” possibilities arises if we allow violations of various energy conditions, such as matter possessing negative energy densities. For example this paper considers “creation of a universe in a laboratory” where from initial configuration of finite size and energy a new expanding semi-closed world forms (see figure for schematic representation of purely spatial geometry at a fixed moment in time),

which could then, in principle enter its own inflationary epoch and Big Bang phase producing a lot of matter content inside this new baby universe, while from the point of view of outside region this system would still have finite size and mass. Note, that the model considered  there (so called galileon field) does not have serious pathologies such as ghosts, superluminal signals etc.
