Classical theism is not a monolithic thing, it comes in many shapes and sizes; typically that is associated with the period classical theism is developed within and associated with. Most notably though for Western Christians, particularly us Protestants in the Reformed tradition, classical theism takes much of its shape from Thomas Aquinas. I thought it would be instructive to see how Bruce McCormack defines classical theism; he writes:
Classical theism presupposes a very robust Creator-creature distinction. God’s being is understood to be complete in itself with or without the world, which means that the being of God is “wholly other” than the being of the world. Moreover, God’s being is characterized by what we might think of as a “static” or unchanging perfection. All that God is, he is changelessly. Nothing that happens in the world can affect God on the level of his being. He is what he is regardless of what takes place—and necessarily so, since any change in a perfect being could be only in the direction of imperfection. Affectivity in God, if it is affirmed at all, is restricted to dispositional states which have no ontological significance.
I think it safe to say that even Karl Barth could fit his understanding of God into this definition provided by McCormack, albeit with qualification. Someone who could not fit into the classical tradition would be Moltmann and the death of God theologians.
Anyway, just a random blog post about classical theism. There are good ways to appropriate and engage with it, and there are bad ways. And of course there are some readings of Barth that would repudiate the hard metaphysicalism in the definition provided by McCormack. In other words some interpretations of Barth don’t allow for the type of metaphysical antecedent theology (like Divine aseity) that classical theism entails. So that is not an uncontroversial point, at least in North America, when it comes to reading Barth.
 Bruce L. McCormack, ed., Engaging the Doctrine of God: Contemporary Protestant Perspectives (Grand Rapids, Michigan: Baker Academic, 2008), 186-87.