Augustine says No, for the following reason: “Ecce, Dominum esse non sempiternum habet, ne cogamur etiam sempiternam creaturam dicere, quia ille sempiterne non dominaretur, nisi etaim ista famularetur” (De Trin. V.16.17).
Translated: “Look, he can’t be ‘Lord’ eternally, lest we also be compelled to call the creature eternal, because he could not eternally lord it, unless the creature also served eternally.”
We could answer this question a bit differently by appealing to the doctrine of election, and say that God is eternally Lord insofar as he has eternally determined to be God in no other way than as the Lord of creation (this is Barth’s approach, I think). But the conceptual distinction is important, nonetheless: to say that God freely chooses to take on the relation of Lord to his creatures is far different from saying that being the Lord of his creatures belongs necessarily to the Godhead; the former preserves both God’s freedom and creation’s gratuity; the latter leaves us with a mere god, bound with his works into an order of totality within which he is a mere artifact.