Say you take a parcel of air and place it inside a rigid container, like a glass bottle. Then put that bottle on a rocket sled and accelerate it to very high speed. Since the air inside the bottle now has a lot of kinetic energy will the pressure inside the bottle change?
There are two answers to your question, as follows. We'll assume the long axis of the bottle is aligned with the direction of the sled's travel.
1) While the bottle on the sled is accelerating, it is as if the bottle is sitting in a powerful gravitational field where "up" points in the direction of travel. The gas molecules then get compacted down into the "down" end of the bottle, leaving fewer of them in the "up" end. You then get a density gradient along the length of the bottle and if you put a series of pressure gauges along the length of the bottle, you'd find more pressure in the "down" end than in the "up" end.
2) When the motor on the sled burns out and stops producing thrust, the gas molecules are then coasting along freely and redistribute themselves evenly along the length of the bottle, the density gradient disappears, and all the pressure gauges will read the same- as if the bottle were not moving. The kinetic energy of their collective motion does not affect their random vibrations, which give rise to the pressure (and temperature) inside the bottle.
This holds true for sled speeds much smaller than the speed of light. Different reasoning must be applied in that case, which I invite the experts here to furnish.