This has some pretty bizarre consequences. If you could maximize the entropy in the system, temperature becomes discontinuous—it jumps from positive to negative infinity.

M. Tazerout, O. Le Corre, S. Rousseau, TDC Determination in IC Engines Based on the Thermodynamic Analysis of the Temperature-Entropy Diagram, SAE Transactions, Vol. 108, Section 4: JOURNAL OF FUELS ...