Thermodynamic classifications and dilatonic black holes

We analyze the thermodynamics of systems which have entropy functions of the type S(m) = am(beta) + b, where m is an extensive variable and a, b, and beta are constants. Such functions apply to dilatonic black holes whose mass is m. This analysis continues our earlier treatment of the general classification of the thermodynamics of systems by whether they exhibit entropy functions which may or may not be either superadditive, homogeneous or concave in the extensive variables on which the entropy depends. This leads to a classification into 8 types of thermodynamics (with several subtypes). We show that only five of these are available for systems having the entropy given above, and these are in fact realized if the constants are given appropriate values.