Q: What is the entropy of nothing?

Mathematician: In physics, entropy relates to the number of states that a system can be in. If a system actually contained absolutely nothing, then (quantum mechanical considerations aside), it would only have one state, and therefore would have 0 entropy (there would be no uncertainty at all about what state it is in). In reality though, things are more complex. Particles and anti-particles will pop into existence in pairs in empty space.