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• $E_\infty$ can be infinite for finite patterns if they grow without bound, as plenty of patterns in CGoL do. (This alone isn't a particularly awful feature, as things go, but it's worth noting for completeness.)

• $E_\infty$ can be undefined if the long term average of $E$ never converges to a (finite or infinite) limit. In particular, I believe there are sawtooth patterns whose long-term average live cell count never converges.

• As noted above, the value of $E_\infty$ (or even whether the limit exists or not) can be computationally undecidable: it should be possible to construct a (family of) pattern(s) encoding an arbitrary Turing machine and its input, such that $E_\infty$ for the resulting pattern depends on whether or not the encoded Turing machine ever halts.

• $E_\infty$ can be infinite for finite patterns if they grow without bound, as plenty of patterns in CGoL do. (This alone isn't a particularly awful feature, as things go, but it's worth noting for completeness.)

• $E_\infty$ can be undefined if the long term average of $E$ never converges to a (finite or infinite) limit. In particular, I believe there are sawtooth patterns whose long-term average live cell count never converges.

• As noted above, the value of $E_\infty$ (or even whether the limit exists or not) can be computationally undecidable: it should be possible to construct a (family of) pattern(s) encoding an arbitrary Turing machine and its input, such that $E_\infty$ for the resulting pattern depends on whether or not the encoded Turing machine ever halts.

• Also, while $E_\infty$ is additive for never-interacting patterns as defined above, it is not an "additive conserved quantity of range $\alpha$" as defined e.g. by Hattori & Takesue (1991). In particular, the "patterns will never interact" requirement for additivity means that $E_\infty$ cannot be defined as a sum of "local energies" for each cell on the lattice, where the "local energy" of a cell only depends on the states of a bounded number of nearby cells.

That said, if that's not a problem for you, it is in fact possible to define an additive conserved quantity for Conway's Game of Life (or, indeed, for any cellular automaton or similar system) that satisfies all the properties listed in your question:

Theorem: Let $E: \{0,1\}^{\mathbb Z^2} \to \mathbb R \cup \{-\infty, +\infty\}$ be an arbitrary additive function assigning an "instant energy" to a particular lattice state (for example, $E(L)$ could simply count the number of live cells on the lattice $L$) and define $$E_\infty(L) = \lim_{n \to \infty} \frac1n \sum_{k=0}^n E(S^{(k)}(L)),$$ where $S^{(k)}(L)$ denotes the lattice state obtained by evolving the lattice state $L$ by $k$ generations under the CGoL rule. Then $E_\infty$, where defined, is an additive conserved quantity. Specifically:

• By definition, $E_\infty$ is conserved, as it only depends on the long term average limit of $E$ as time tends to infinity. In particular, it's not hard to show that if $E_\infty(L)$ is defined, then $E_\infty(S^{(k)}(L)) = E_\infty(L)$ for any finite $k$.

• If $E$ is additive for non-interacting patterns, then $E_\infty$ is also additive for patterns that never interact as they evolve. In particular, if the operator $\oplus$ denotes some method of merging two lattice states into one, such that $E(A \oplus B) = E(A) + E(B)$ whenever both expressions are well defined, and if $L_1$ and $L_2$ are two lattice states such that $S^{(k)}(L_1 \oplus L_2) = S^{(k)}(L_1) \oplus S^{(k)}(L_2)$ for all $k \ge 0$ (i.e. if the evolutions of $L_1$ and $L_2$ do not interact when combined using $\oplus$), then $E_\infty(L_1 \oplus L_2) = E_\infty(L_1) + E_\infty(L_2)$.

The price to pay for these seemingly convenient properties is that the "eventual average energy" functional $E_\infty$ also has a couple of awkward features:

• $E_\infty$ can be infinite for finite patterns if they grow without bound, as plenty of patterns in CGoL do. (This alone isn't a particularly awful feature, as things go, but it's worth noting for completeness.)

• $E_\infty$ can be undefined if the long term average of $E$ never converges to a (finite or infinite) limit. In particular, I believe there are sawtooth patterns whose long-term average live cell count never converges.

• As noted above, the value of $E_\infty$ (or even whether the limit exists or not) can be computationally undecidable: it should be possible to construct a (family of) pattern(s) encoding an arbitrary Turing machine and its input, such that $E_\infty$ for the resulting pattern depends on whether or not the encoded Turing machine ever halts.

Is this definition useful for anything? I'm not sure. On one hand, it does let you assign an "energy" to simple patterns like still lifes, oscillators and spaceships, and have it be additive as long as those patterns never interact. You can even assign an energy to any random soup that eventually decays to a finite amount of non-interacting ash. On the other hand, there's no general way to determine the energy of an arbitrary pattern (or even to determine whether it's defined or not) other than by simulating it until it settles into a collection of non-interacting parts whose population growth is predictable. Some patterns never do, and for some you may never be able to tell if they do or not.

Conserved quantities like energy are characteristic of time-reversible dynamical systems. Conway's Game of Life is a dissipative, non-time-reversible system, and thus not likely to have any nontrivial conservation laws.

In fact, I can't even think of any invariant boolean properties preserved by the CGoL update rule:

• "There are live cells" is not an invariant property, since patterns can die.
• "There are finitely many live cells" is not an invariant property, since even infinite patterns can die.
• "There are exactly four live cells arranged in a 2x2 block" is not an invariant property, since other patterns can turn into a block.
• "There is no Garden of Eden pattern" is not an invariant, since a GoE can be present in the first generation (but will never be present in any later one).
• "There is no orphan pattern" is also not an invariant, since even orphans can die if disrupted from the outside.

In principle, a cellular automaton rule could have non-trivial invariants, such as a "superstable orphan" pattern that can neither be formed unless it is already present on the lattice nor destroyed if it is. But as far as I know, no such patterns exist in Conway's Game of Life (nor would they seem to be quite what you're looking for, even if they did exist).

There are, of course, trivial invariants based on the eventual evolution of the pattern under the iterated CGoL update rule, such as "this pattern will eventually die." (Incidentally, I'm also pretty sure that this property, and others like it, are computationally undecidable — it should be possible to construct a pattern in CGoL that will simulate a Turing machine and die out if and only if the machine halts.) But a conserved property that can only be calculated by simulating the system for an unbounded number of steps is hardly useful for doing physics with.

Conserved quantities like energy are characteristic of time-reversible dynamical systems. Conway's Game of Life is a dissipative, non-time-reversible system, and thus not likely to have any nontrivial conservation laws.

In fact, I can't even think of any nontrivial invariant boolean properties preserved by the CGoL update rule:

• "There are live cells" is not an invariant property, since patterns can die.
• "There are finitely many live cells" is not an invariant property, since even infinite patterns can die.
• "There are exactly four live cells arranged in a 2x2 block" is not an invariant property, since other patterns can turn into a block.
• "This pattern oscillates with a period of $n$ steps" is not an invariant property, since non-oscillating patterns can evolve into oscillators.
• "There is no Garden of Eden pattern" is not an invariant, since a GoE can be present in the first generation (but will never be present in any later one).
• "There is no orphan pattern" is also not an invariant, since even orphans can die if disrupted from the outside.

In principle, a cellular automaton rule could have non-trivial invariants, such as a "superstable orphan" pattern that can neither be formed unless it is already present on the lattice nor destroyed if it is. But as far as I know, no such patterns exist in Conway's Game of Life (nor would they seem to be quite what you're looking for, even if they did exist).

There are, of course, trivial invariants based on the eventual evolution of the pattern under the iterated CGoL update rule, such as "this pattern will eventually die." (Incidentally, I'm also pretty sure that this property, and others like it, are computationally undecidable — it should be possible to construct a pattern in CGoL that will simulate a Turing machine and die out if and only if the machine halts.) But a conserved property that can only be calculated by simulating the system for an unbounded number of steps is hardly useful for doing physics with.