Thermodynamics really just say that it must increase or stay the same. The second law of thermodynamics was originally formulated as an inequality (the Clausius Inequality). It's not a question of speed of increase; just that in a macroscopic and isolated system, the entropy cannot decrease.
I can say, however, that in thermodynamic equilibrium the entropy does not increase. It simply stays the same. This is more of a definition, though.
