**Q:** *Does the temporal translation symmetry of Conway's universe give rise to a conserved quantity, that we might be able to call an "energy"?*

As <A HREF="https://ui.adsabs.harvard.edu/abs/1989Natur.342..780B/abstract">noticed</A> in the earliest studies of Conway's Game of Life, it has no local conservation law --- it is not possible to define a locally conserved energy functional. The dynamics does have temporal translation symmetry, but Noether's theorem (which ties a symmetry to a conservation law) does not apply because the dynamics is discretized in space and time. Noether's theorem requires that the dynamics is obtained from a differentiable Lagrangian, which the Game of Life lacks.