The synthesist’s eidolon isn’t a separate creature, not really, not in game terms. It doesn’t have its own space, AC, saves, and so on. It doesn’t get targeted separately from the synthesist. Instead, it’s more like a buff effect that’s operating on the synthesist, albeit a strange one.

Which means not only does the eidolon not lose hp, *it doesn’t even **have** hp to lose*. It’s not a creature with hp. It’s a buff that grants temporary hp, and has the unusual condition that it ends when those temporary hp are lost.