Does localized energy operator make sense in quantum field theory?

In quantum field theory in weak gravity regime, it is possible to trace out some region to obtain reduced density matrix of states restricted to hypersurface $V$.

I believe the same idea carries to energy operator, though I am not sure whether this really is the case, given that no one really has discussed/has used such localized energy operator.

Given that Hamiltonian is integration of Hamiltonian density at spacetime point, it seems natural that tracing out some region would be possible for energy operator – though energy operator would not be Hamiltonian in usual sense. Would this be wrong?

I do get that Hamiltonian is usually defined as generator of unitary time translation. But surely, there is sense that energy in the conventional sense other than time translation generator would mean something? And I definitely do not mean modular Hamiltonian by energy operator.