Does a Sunfire Cape still hurt enemies while you are not in an 'active' state?

Does the Sunfire Cape still apply AoE damage for the few seconds you are golden from Zhonya's Hourglass, or during the revive of a Guardian Angel?

Yes. Any aura from passive objects persists when using an item such as Zhonya.

People discussed it in this thread:

All non-channel abilities, such as Kennen's or Fiddlestick's ult, or passive auras such as Sunfire Cape, Runic Bulwark or Frozen Heart will persist through the duration of Zhonya's.

The effect is also active when your player is down with the effect of Guardian Angel. Players in this thread also discussed about this point.

I was in a team battle with a rammus who had both items and we killed him and he was being revived by guardian angel and I was low on health and suddently I die and it was to the sunfire cape of the dead rammus.

Even if those threads are old, it seems those effects didn't change nowadays.