Magnetothermodynamics is the study of compression and expansion of magnetized plasma with an eye toward identifying equations of state (EOSs) for magneto-inertial fusion experiments. We present recent results from Swarthmore Spheromak Experiment (SSX) experiments on the thermodynamics of compressed magnetized plasmas called Taylor states. In these experiments, we generate twisted flux ropes of magnetized, relaxed plasma accelerated from one end of a 1.5-m-long copper flux conserver and observe their compression in a closed conducting boundary installed at the other end. Plasma parameters are measured during compression. The instances of ion heating during compression are identified by constructing a pressure-volume diagram using measured density, temperature, and volume of the magnetized plasma. While we only measure compression up to 30%, we speculate that if higher compression ratios could be achieved, the compressed Taylor states could form the basis of a new kind of fusion engine. The theoretically predicted magnetohydrodynamic (MHD) and double-adiabatic [Chew-Goldberger-Low (CGL)] EOSs are compared to experimental measurements to estimate the adiabatic nature of the compressed plasma. Since our magnetized plasmas relax to an equilibrium described by MHD, one might expect their thermodynamics to be governed by the corresponding EOS. However, we find that the MHD EOS is not supported by our data. Our results are more consistent with the parallel CGL EOS suggesting that these weakly collisional plasmas have most of their proton energy in the direction parallel to the magnetic field.