Yes, but because space is expanding - i.e. there's a value of meters per meter per second for the expansion of space, then every point in space has a sphere around it of locations which are sufficiently far away that they are growing more distant at the speed of light.
Any virtual particle pair which appears along that horizon can potentially be caught on the wrong side of it - i.e. one virtual particle is in a part of space which is now far enough way to be expanding faster then light, whereas the other particle is at a location close enough that it is not.
At the moment that happens, there's a particle or photon which is now inside the light cone of a distant object, paired with a particle that it will never meet again because it's outside of it.
As a result, you get Hawking radiation: because one particle can go off and interact with your universe, but it's partner will never be able to causally effect anything inside that horizon again. So the virtual particle has to become real.
It's a very large horizon, so the Hawking radiation is very weak (it's always a weak effect). We can't see it because the cosmic microwave drowns it out.
Broadly Hawking radiation intensity goes with curvature - i.e. a more curved surface has a better chance of separating a particle pair then a flatter one. This is because a sharper curve means more vectors which carry you away from the event horizon.
This is also why small black holes evaporate faster then big ones - as the black hole shrinks, the Hawking radiation intensity increases because it's curving more and more (hence why microscopic black holes don't devour everything).
Any virtual particle pair which appears along that horizon can potentially be caught on the wrong side of it - i.e. one virtual particle is in a part of space which is now far enough way to be expanding faster then light, whereas the other particle is at a location close enough that it is not.
At the moment that happens, there's a particle or photon which is now inside the light cone of a distant object, paired with a particle that it will never meet again because it's outside of it.
As a result, you get Hawking radiation: because one particle can go off and interact with your universe, but it's partner will never be able to causally effect anything inside that horizon again. So the virtual particle has to become real.