The traditional Christian view that God is unchanging has been accused of being a fruit of Greek ideals of perfection (and what's wrong with that?). Here I want to motivate this view by thinking about our mental life.
But our conscious states are divided between times in much the way that the two centers of consciousness of a split-brain patient are divided from each other. My present state of consciousness only includes shadowy reminders of what I was aware of five minutes ago and vague premonitions of what I am about to be aware of. My temporality makes me like a patient split into untold numbers of centers of consciousness associated with different times (perhaps in a continuous way, with overlapping between close-by centers, since many of our mental states themselves persist over short amounts of time). We are deeply internally disunited--our "transcendental unity of apperception" is quite limited. Such deep internal division and disunion is surely not what the perfect being would experience (at least not in his proper nature—an Incarnation might make for such an experience, and the above reflection should make us grateful that he took up this deeply divided existence for our sake). This is not a matter of some "Greek ideal" of perfection. It is simply the intuition that mental division within oneself is an imperfection.
The above argument presupposes eternalism. But presentism only introduces even greater limitation in our mental life by making the future and past conscious states not be ours.
So we have good reason to think of God's mental life as all-encompassing, of God living an infinitely rich mental life all at once, as Boethius said. But God is a mind and surely all of his mental states are conscious. This gives us good reason to think God is unchanging.