Black holes emit Hawking radiation regardless of ambient temperature. It's just that right now, the mass(-energy) of photons falling into the hole from the CMBR is outpacing the mass loss to Hawking radiation for large black holes, so large black holes aren't currently evaporating from their own Hawking radiation.

It's like standing next to a campfire. Your body is still emitting thermal radiation in the IR, it's just that you're receiving more thermal radiation from the fire than you're putting out.