One thing I haven't seen talked about at all: how quickly would space heat up?

I presume Earth's gravity largely keeps the exosphere it has around it. With some modest fractional % lost year by year. There is a colossal vast volume out there! But given that there's so little matter up in space, what if any temperature rise would we expect from say a constant 1TW of heat being added?

The sun’s radiation hitting earth is 44,000 terawatts. I think we’re fine with an “extra” terawatt. (It’s not even extra, because it would be derived from the sun’s existing energy.)

https://www.nasa.gov/wp-content/uploads/2015/03/135642main_b...