One aspect of entropy that I always find counterintuitive is that unlike mass, charge, etc. it is not a physical quantity. In fact, from the point of view of an experimenter with perfect information about a physical system, the entropy of the system is exactly conserved over time (as made precise by Liouville's Theorem). The Second Law survives in this setting only in the most trivial sense that a constant function does not decrease.
It's only when you start making crude measurements---lumping positions into pixels, clouds of particles each with their own kinetic energy into a single scalar called "temperature," etc---that you start to see a nontrivial entropy and Second Law. Different ways of lumping microstates into macrostates will give you different (and inconsistent) notions of entropy.
It's only when you start making crude measurements---lumping positions into pixels, clouds of particles each with their own kinetic energy into a single scalar called "temperature," etc---that you start to see a nontrivial entropy and Second Law. Different ways of lumping microstates into macrostates will give you different (and inconsistent) notions of entropy.