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The first study of the game of life with random initial conditions that I could find was this paper:

F. Bagnoli, R. Rechtman and S. Ruffo, Some Facts of Life, Physica A 171, 249 (1991) doi:10.1016/0378-4371(91)90277-J.

In it they attempt a kind of mean-field analysis (which I haven't digested) as well as do some numerical experiments which suggest that the system approaches a nontrivial asymptotic density for all $0\lt p\lt 1$.

Here's a plot from their paper of results on 256 by 256 toruses (their $\rho_0=p$):

They also investigated the temporal behavior, however I found more detailed simulations and discussion of scaling in this paper:

J. B. C. Garcia, M. A. F. Gomes, T. I. Jyh, T. I. Ren, and T. R. M. Sales. Nonlinear dynamics of the cellular-automaton ‘‘game of Life’’. Phys. Rev. E 48, 3345–3351 (1993)

Here is a plot showing the average number of connected clusters of live sites versus time.

They summarize their findings as follows (below $\varphi$ represents some quantity like the number of live clusters or the total number of live sites, etc.):

For initial occupation probabilities satisfying $0.15\leq p \lt 0.75$, each one of the different statistical functions $\varphi$ describing the dynamics of the GL may be divided in general in three intervals: First, a region extending from $t =0$ to $t\simeq L^{1/2}$ presenting large fluctuations in $\varphi$; second, a scaling region characterized by a power-law dependence between $\varphi$ and $t$, from $t\sim L^{1/2}$ to $t\sim L^{4/3}$ and finally the "steady state" or stabilization region (the SOC [self-organized criticality] state of Bak, Chen, and Creutz [4]) extending from $t\sim L^{4/3}$ to infinite and characterized by small fluctuations of $\varphi$ around some average value $\varphi_0$. These results are obtained from extensive numerical simulations on lattices with different values of L. The critical exponents obtained in the scaling region are robust and do not depend on $p$, for $0.15\leq p \lt0.75$. [...] For low initial occupation ($p\lt 0.15$) or high density occupation ($p \geq 0.75$) the scaling region disappears, and now the domain for large fluctuations of $\varphi$ extends from $t =0$ to $t\sim L^{4/3}$. This region is followed by the "steady state" characterized by small fluctuations around an average value.

I didn't see any rigorous arguments in the papers I found but that doesn't mean there aren't any out there. In particular as Gil hints there is a strong connection to bootstrap percolation, and perhaps some of the theorems proven there apply here as well.