Does the universe depend on God for its existence? To most people, the answer depends upon whether the universe began to exist or not. Today, we have multitudes of scientific theories either attempting to explain the possible origin of our universe or attempting to demonstrate that the universe has always existed. The notion that the right answer lies in whether the universe began to exist or not, though, is thoroughly modern.
Aristotle could hold simultaneously that the universe has always existed and that it depends upon God for its existence. How?
I am sitting on a chair as I write this. Let us assume that I have been sitting on a chair and will continue to sit on a chair eternally? Do I still need the chair to hold me up? If I do need it, we have a notion of causation which does not rely on temporal duration in order to exist.
We can see this again in another example. At every moment I am depending upon many things in order to remain alive. I need oxygen to breathe, and in order for oxygen to exist on Earth I need the Earth to have the same atmosphere it has right now, and for the Earth's atmosphere to remain the same, let's say that it needs to remain a certain distance from the Sun, and for this to stay the same the Sun cannot change in any drastic way, and for this to happen let us suppose a million different things in the universe have to remain as they have always been.
This entire chain of dependency does not need to stretch back in time in order to have real power. At every moment of my existence all of these factors must be what they are. If one link was altered, then everything after it would be altered too.
What we have, then, is an atemporal notion of causation. In other words, we can say that the universe has existed forever, just like I have always been sitting on my chair. We can also say, though, that the universe depends on God for its perpetual existence, just like I still need a chair in order to remain seated even if I have been sitting always.
Any philosopher who does not acknowledge this when discussing God's Existence is unworthy of your attention. Any book which does not acknowledge this can be shut.