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 noticed 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.

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 noticed 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 SmoothLife would not be sufficient to apply Noether's theorem.