[OscarCunningham]

I don't think temperature would be measured as energy, but rather as energy per information; e.g. joules per bit. Boltzmann's constant defines the degree as one joule per a very large number of bits, to get numbers convenient at macroscopic scales.

[kgwgk]

The argument of the exponential E/kT needs to be dimensionless. If E and T are not homogeneous you will still need a k to “cancel” the units.