I think the literal physical stars are referred to. The centaur also thought the stars would go out:

"So the wandless have become wiser than the wizards. What a joke! Tell me, son of Lily, do the Muggles in their wisdom say that soon the skies will be empty?"

"Empty?" Harry said. "Er... no?"

"The other centaurs in this forest have stayed from your presence, for we are sworn not to set ourselves against the heavens' course. Because, in becoming entangled in your fate, we might become less innocent in what is to come. I alone have dared approach you."

"I... don't understand."

"No. You are innocent, as the stars say. And to slay something innocent to save oneself, that is a terrible deed. One would live only a cursed life, a half-life, from that day. For any centaur would surely be cast out, if he slew a foal."

Literal stars. Literally torn apart. The sun must be tamed. And more distant stars are also dangerous.