In the top answer to this question (https://physics.stackexchange.com/questions/231017/is-the-boltzmann-constant-really-that-important/231065) I read that the Boltzmann constant is just a dummy factor which converts energy to temperature.

But that allows me to put another numerical value in place of the Boltzmann constant but keep the dimension J/K.

I.e. what if, in
$$S=c\ln W,$$
I put $c\approx 56  \, \mathrm J/\mathrm K$ in place of $c\approx 1.38\cdot 10^{-23} \,  \mathrm J/\mathrm K$?

On the page [Thermodynamic beta][1], the Boltzmann entropy using the Boltzmann constant implies thermodynamic beta, which implies (according to [Derivation of Boltzmann Distribution Law][2]) the Boltzmann distribution.

So the Boltzmann distribution depends on the numerical value of Boltzmann constant. Then why is the Boltzmann constant just a dummy factor?


  [1]: https://en.wikipedia.org/wiki/Thermodynamic_beta
  [2]: https://bouman.chem.georgetown.edu/S98/boltzmann/boltzmann.htm