Normally, the onus is on one who seeks to demonstrate the truth of claim, not its negation. Thus, the onus is on the one who seeks to show the universe is aware, not that it isn't.
However, it is a property of systems that possess some sense of self-awareness/consciousness (even those systems generally not regarded as being sufficiently aware as to be considered conscious) that the emergence of consciousness be unaware of the components of the system that produce it (the CNS, the brain, the PFC, or however one wishes to partition the nervous system into "responsible for consciousness" vs. "not responsible for consciousness").
Put simply, it is almost universally believed that human consciousness is not possible without the brain (even many dualists and believers in the soul hold that, as living humans, our "souls" or "minds" or whatever "use" a physical brain- the infamous "ghost in the machine). Thus without neural interactions we cannot be conscious (or, for those who believe in souls and the existence of "self" after death, consciousness must emerge from something other than the brain).
If the universe were conscious, than the universe would necessarily be ignorant of the components of the system whence this consciousness emerged. The "universe" wouldn't be conscious of its "brains'" parts/"neurons". However, the components of the universe consist of everything that exists. For the universe to be conscious, to have self-awareness, it can't be conscious of the components that allow it to be conscious any more than we can be conscious of our neurons. But nothing else exists for the universe to be conscious of.
It's like a brain being conscious in vacuum. How can it be conscious (and of what) when all that exists is that which would make it conscious if there were anything to be conscious of?