Could you just throw the heat into Vesper?
Vesper, the Vigil's black hole, glows at only 4.3 billionths of a kelvin. Nothing else in the Vigil's sky is colder. The Vigil's computers run at 300 K and their radiators at 30 K, and heat flows from warm to cold, so the idea is irresistible: point the waste heat at the hole and let it fall in. Details of that glow are on the page about why black holes glow.
Guess first The Vigil makes a petawatt of waste heat. Roughly what share of it could Vesper swallow if the radiators faced it?
Almost none. A black hole is a very small target, and the heat from a 30 K radiator is faint, spread-out light. Vesper can take at most 4.9 × 109 watts of it, which is 4.9 × 10−6 of the petawatt. Nearly all of the heat still has to leave the old-fashioned way.
Here is the catch. A hole only swallows light that passes inside its capture radius, about 92 km from the centre for a hole of Vesper's mass (worked out without its spin, as are the limits below). Mirrors and lenses can steer light, but they cannot make it brighter than its source, so you cannot squeeze faint, wide-angle heat through a small window, any more than a magnifying glass can make sunlight hotter than the surface of the Sun. That rule about brightness caps what the hole can take: no more than a ball the size of its capture radius would soak up if it sat bathed on every side in 30 K glow. That comes to 4.9 × 109 watts, and no arrangement of mirrors can beat it.
To push the whole petawatt through the same window, the heat would have to arrive as light at 638 K. That is hotter than the 300 K computers that made it. You would have to spend work pumping the heat uphill before the coldest thing around could take it, which defeats the idea.
Vesper's small capture area prevents it from absorbing a significant share of the waste heat. The Vigil therefore needs large radiators that emit into space. The full page on the Hawking sink derives the capacity limit.
The universe keeps cooling, and in 351 billion years the sky will be colder than Vesper. Thinking in a cooling universe follows the change in relative temperatures.