Canonical ensemble theory is used to obtain simple exact scaling relationships (homology transformations) which establish equivalences between certain thermodynamic quantities for two state weak valued and mean valued energy quantum systems that are described by the same Hamiltonian. These equivalences define how a thermodynamic quantity for a weak (mean) valued energy system can be represented by an appropriately scaled version of the same quantity for a mean (weak) valued energy system. This provides: (i) a method for quantifying the thermodynamic properties of the typically more complicated (and difficult to prepare) weak valued energy systems in terms of those of the generally less complicated mean valued energy systems; and (ii) a systematic way to compare the properties of weak and mean valued energy systems. The results are validated and illustrated numerically by applying them to a spin ½ system in a uniform magnetic field. Several apparent thermodynamic effects induced by the weak measurement and state post-selection process required to prepare a weak valued energy system are also identified for this system.