The Book of Life causes a lot of problems for the open theist.
Here is what I am taught.
Every name of every human being born was written in the Book of Life before the foundation of the world. When a human being dies his or her name is blotted out if they did not believe. When the very last human being dies, and the very last unbeliever is blotted out, then the “evolving” Book of Life will become the Lamb’s Book of Life.
Every reference to names being written in the Book of Life are before the foundations of the world. In Psalms 69 we have David lamenting for his unbelieving enemies to be blotted out of the Book of Life while his enemies were still alive.
The problem with the open theists view (Lighthouses’) of names being written into the Book of Life upon regeneration is this would mean that every baby that dies would go to hell. The only way a person’s name is blotted out of the evolving Book of Life is for unbelief and unbelief only. Sins have nothing to do with the blotting out. Since babies cannot reach the age of accountability there names are not blotted out when they die, and thus end up in the Lamb’s Book of Life.
Now, the only way God could know the names of every human being ever to be born is by exhaustive foreknowledge. And since God is outside of time, God even knows who is in the Lamb’s Book of Life, and who is not. Thus the people in the Lamb’s Book of Life are the elect, and predestined with blessings and provisions.
The only way the open theist can fit any Book of Life into open theism is to suggest as Lighthouse did, that names are entered at salvation. Again, this not only conflicts with every mention in the Bible of the names being entered before the foundation of the world, but would also send babies to hell.
Remember, from the perspective of time, the Book of Life is an evolving book and salvation is available to all. Since God is outside of time, He not only could enter every name in the evolving Book of Life before Adam, but He also knows who ends up in the Lamb’s Book of Life.