Yes, God is immutable (unchanging through time). God-Consciousness is and always will be God-Consciousness. God Himself is unchanging in that He has no need to change in His formless state. With His presence in the Eternal Now, there is no need to change in essence. Nevertheless, from the viewpoint of His Creation, there is all that change, growth, evolution, experience, invention, and development offers. God-Consciousness experiences movement, change, life, joy, and excitement through our Higher Selves. This is within His overall immutability.
Experience does not change God-Consciousness in essence as He has no need to change. Experience is experience and that is all it is. God is beyond time and is already conscious of past, present and future without taking the experience of free will away from His Creation.
Taking mathematical laws as an analogy, we can say that 1 + 1 = 2 is an immutable law. In fact all mathematical and physical laws are unchanging, assuming of course that they have been correctly formulated in the first place. If they have been incorrectly stated, all we need to do is correct the error. It is not that the law was wrong in itself or that the law was not immutable, but merely that Man made an error in its expression.
So, God-Consciousness is like the immutable mathematical law 4 + 3 = 7, and His experience (in time) being the figuring out of the value of a in the equation,
The mathematical law is immutable, but in time there is the experience of solving the equation. In other words, having fun and experience within time.
As we have said, we cannot break physical and mathematical laws. They are what they are as immutable facts, and we live within those constraints. In truth, we would not be alive unless those constraints were in place. The universe only works as it does because of those immutable physical and mathematical laws.
God, so far beyond His Creation that He is really beyond our comprehension and description, can nevertheless be accepted as being as immutable as His laws.
(Parts of this blog are quoted from my book, The God Franchise)