do relativistic velocities change the apparent entropy content of the moving object? Imagine a box of hot gas. It has a certain (large) amount of entropy, which we can relate to the amount of information needed to completely specify the position and velocity of every gas particle in the box. Attach a clock and rocket motor to the box and accellerate it to close to the speed of light. As the rocket zooms past us, we notice that the hands on the clock it carries are seemingly frozen, and with some special apparatus we notice that the gas particles themselves do not seem to be moving either. Furthermore, the length of the box in the direction of travel has apparently vanished, rendering the box two-dimensional. The amount of information needed to completely specify the contents of the box is reduced, because 1) the box is missing one degree of freedom it had while at rest (relative to us) and 2) every one of the gas particles now has exactly the same velocity, say .99999c in the direction of travel, and their positions within the box itself are no longer changing with time. It would seem then that the box as observed from a nonmoving reference frame now contains less entropy than it did while at rest in that frame. (An observer attached to the box would detect no such changes in it.) What is wrong with my understanding of this?
 A: 
and with some special apparatus we notice that the gas particles themselves do not seem to be moving either.

The assumptions imply the particles move with high speed. Their velocities differ less one from each other, but the particles are moving.

The amount of information needed to completely specify the contents of the box is reduced, because 1) the box is missing one degree of freedom it had while at rest (relative to us) and 2) every one of the gas particles now has exactly the same velocity, say .99999c in the direction of travel, and their positions within the box itself are no longer changing with time.

Each of these statements is false. You are neglecting the difference between the transformed quantities based on the result this difference is very small (i.e. you are truncating the numbers). This is not justified if we want to "completely specify the contents of the box."
If the numbers are not truncated, the Lorentz transformation is reversible, so specification of state of the system requires the same amount of data in any inertial reference frame.
