**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 firstly because the dynamics is discretized in space and time, and secondly because the dynamics is not based on a Lagrangian. So even a generalization along the lines of <A HREF="https://arxiv.org/abs/1111.1567">SmoothLife</A> would not be sufficient to apply Noether's theorem.