Is our universe stable or metastable?
Yes ... and no.
Experimentally, there's no indication that the laws of physics are changing (c.f. conservation of energy: if the laws of physics change with time, energy would not be conserved via Noether's theorem). So if you're an empiricist, you'd conclude that the universe is stable (or at least its mean lifetime must be more than ~14 billion years, the age of the universe).
However, if we believe string theory (which has no compelling experimental evidence), then we expect roughly $10^{500}$ &quot;vacuum states&quot;, which correspond to $10^{500}$ different possible universes, each with their own laws of physics. It's possible one of these other universes have a lower &quot;total energy&quot; than us (this word's meaning is vacuous since if one varies the laws of physics, the word 'energy' itself might lose meaning). This means it's in principle possible that our universe will quantum tunnel to a universe at lower energy, in which case our universe is metastable.
Of course, this leads to other problems. How would one prove the multiverse exists? If a generally accepted way exists, I've not heard of it. But if you can't prove the multiverse exists, how would you prove our universe is metastable? For that matter, how do we know the electron is stable? Why can't it have a mean lifetime of, say, $10^{100}$ years (so long that we don't observe it decaying)?
At this point, we're purely in the realm of speculation. We're talking about things so far above the experimental threshold that some people will just dismiss it as pointless. Suffice to say that if the universe isn't stable and quantum tunnels to a lower energy, we won't be able to see the tunneling coming, and we're all instantly dead anyway. Whether the question is worth thinking about further depends on the individual.
tl; dr:

Experimentally, universe is stable.
Theoretically, maybe it isn't, depending on what your theory is because we don't have a widely-accepted, experimentally-backed theory.