Any civilization that evolves during its sun’s main sequence and then finds a way to survive the red giant and supernova stages, will also probably find a way to create a Dyson sphere around the surviving white dwarf. For that reason, these stars may be more likely to host such a structure.

What’s more, a white dwarf is a better host for a Dyson sphere. The habitable zone around a white dwarf is closer to the star, so such a sphere would be smaller. A 1-meter-thick sphere built in the habitable zone around a white dwarf would require some 10^23 kilograms of matter, just a little less than the mass of our moon.

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