Comparing a civilization that consumes 10,000X more energy than we do planet-wide to our technology is comparing apples to coconuts
You (and some of your commenters) are assuming that by the time our civilization advances to Kadashev Type I status that we haven't improved the efficiency with which we consume power. That's a little short-sighted.
A good example of the problem is comparing a simple AM radio from the good old days (say, 1950s) to the transistor radios of my youth (1970s) to the iPod-ish stuff of the late 2000's.
You could literally cook an egg on those old radios (I've had the joy of repairing them. Sing it with me, children! We love radio tuuuuuubes!).
My transistor radio (emblazoned with a lovely red, white, and blue "76" denoting that banner year) fit in the palm of my hand (if not in my pocket) and ran on a 9v battery. By comparison it created no heat at all (although it did a bit).
And the iPod was basically a device an inch square and a quarter of an inch thick using a 1.5v lithium button battery that could play music forever compared to my transistor radio, clear music to boot, and its heat generation was basically nada, zilch, (almost) zero.
Efficiency is one of the hallmarks of technological advancement. The assumption that tomorrow's fusion reactor will generate as much heat-per-kilowatt compared to yesterday's Three Mile Island is ludicrous.
So, how do they keep from boiling their oceans and igniting their atmosphere?
By using power a whole lot more efficiently than we do. We can speculate that this will include things like room-temp superconducting materials, molecule-thin insulators on micron-sized metal windings for motors, much lower conducting resistance and much higher insulating resistance in our semiconductors, etc., but the reality is we have no blooming idea how this will happen (any more than we know how to practically generated 10,000X the energy our planet uses today). But, you didn't (and shouldn't) ask how...
Technological efficiency would increase with all other kinds of technology, allowing the consumption of ever greater amounts of power without burning up the planet.