You are absolutely right about the dimensional analysis. The use of $ \ln T $ etc. is always a shorthand for $ \ln \left(\frac{T}{T_0}\right) $, which is okay to use if for some reason you don't care about $ T_0 $, i.e. because it cancels out or you are interested in the asymptotic behaviour only.

In any expression where you have to take derivatives to get get observable quantities (partition function, generating functional etc.), it's okay to leave off the scale:

$$ \mathrm{d} \ln \left(\frac{T}{T_0}\right) = T^{-1} \mathrm{d}T $$

independent of $ T_0 $.

So: it's a lazy shorthand - the kind of thing much beloved by physicists. :)

You are absolutely right about the dimensional analysis. The use of $ \ln T $ etc. is always a shorthand for $ \ln \left(\frac{T}{T_0}\right) $ which is okay to use if for some reason you don't care about $ T_0 $, i.e. because it cancels out or you are interested in the asymptotic behaviour only.

In any expression where you have to take derivatives to get observable quantities (partition function, generating functional etc.), it's okay to leave off the scale:

$$ \mathrm{d} \ln \left(\frac{T}{T_0}\right) = T^{-1} \mathrm{d}T $$

independent of $ T_0 $.

So: it's a lazy shorthand - the kind of thing much beloved by physicists. :)