There are different temperatures in different zones. When applying the Boltzmann formula we consider only the surface of the core, that in an Ecat SK is about 0.5 cm^2 and is where all the energy id produced.
Just for AA, who does not bother with the tech stuff.
This is a "friendly" question, in the sense that it does not address the key issue with estimating power out as Rossi does with the QX. That issue is Rossi's assuming that his plasma radiation is thermally in equilibrium and therefore obeys Planck (S-B) law.
I'm sure AA, with his engineering experience, will be able to understand that radiation from fluorescent bulbs is forced (not equilibrium) and therefore not thermal in distribution. Its spectral content is very variable.
Here are already several good answers, but one thing hasn't been addressed, which might be what your friend refers to. The Sun and an incandescent bulb both emit (close-to) Planck spectra (as shown in Tomi's and Cort Ammon's answers). In contrast, fluorescent bulbs, or tubes, emit spectra that have multiple spectral lines. Depending on which gas is used in the tube, or what material the tube is coated with, various spectra can be achieved.
Here is a better example; the spectrum from mercury plasma. If measured as rossi does, eyeballing those peaks, it would give an COP multiplier of about 100X!
I'm just here emphasising the final error class in my table, which the QX seems particularly suitable to generating. Rossi has discovered that Lugano-style optical measurement is capable of large inaccuracy and false positives. He has, felicitously, discovered a system where naive optical power measurement gives much more inaccurate results!
Perhaps, AA, you understand why I'm not inclined to believe anything Rossi says?
Scrolling down the screen, below the “Modelling Emission Spectra... “ paper on Research Gate, one finds this little gem: