> To use, not to collect. And thermodynamics says you can't just store it.

I don't think these semantic games are productive. Thermodynamics says you can transform energy. "Collect" in this context means using energy in a way that allows you to retrieve it in the future. For example, charging a battery or condenser with light with a PV panel, powering a motor that accelerates a flywheel, coiling a spring, heating a material, etc.

If you have a Dyson swarm, the best way to "store" the power output of a star for later use is to perform star lifting, making the star itself smaller and lighter so it doesn't burn as hot in the first place, with the extra mass being used to construct a collection of gas giants in the same system.

If this kind of thing appeals to a group with meaningful control over a Dyson swarm, given that Dyson swarms and the capacity to make them can also be used to build and power a rapid, direct, and near-simultaneous colonisation of all galaxies in our future particle horizon (with significant levels of redundancy, though obviously we can't determine if "significant" is "sufficient"), you should anticipate the first such group turning almost every star in the universe into a red dwarf (or brown dwarf, for those they don't care to colonise just yet) in very short order.

I don't know if anyone's been looking for signatures of this specific thing; though I am told that "1 AU sphere of room-temperature metal" would be quite easy to spot, if you downsize the stars first then the size of the corresponding room-temperature sphere gets smaller, and I don't have a quantifiable number for "quite easy" nor any sense of scale for how much effort is going into such searches.

Unless your civilization has ways of infinitely storing energy or exporting it somehow you'll need to be in equilibrium over the long term. This means radiating away the waste heat, at stellar scale.