In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames?
(I.e. In the lab frame $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u = {1 \over 2} m \overline {(v+u)^{2}}$).
The definition of temperature in the kinetic theory of gases emerges from the notion of pressure. Fundamentally, the temperature of a gas comes from the amount, and the strength of the collisions between molecules or atoms of a gas.
The first step considers an (elastic) impact between two particles, and writes $\Delta p = p_{i,x} - p_{f,x} = p_{i,x} - ( - p_{i,x}) = 2\,m\,v_x $ where the direction $x$ denotes the direction of the collision. This, of course, is considering that the two particles have opposing velocities before impact, which is equivalent to viewing the impact in the simplest frame possible.
This calculation is independent of frame translation, as it will add the same velocity component to both velocities, and the previous equation relies only on the difference in velocities.
The second step uses the ideal gas law to get to $T \propto \frac{1}{2}mv^2$.
For more detail you can check this Wikipedia article.
So the invariance with frame translation of the temperature is due to the invariance of pressure, which only considers relative velocities.
Temperature is about incoherent movement of particles all moving in random directions. When the gas in car engine explodes and the temperature and pressure increases, this random motion of gas particles that we call heat, is converted to the coherent linear motion of the particles in the piston all moving in the same direction. Coherent motion is not the same as temperature.
When we move relative to contained volume of gas with a given temperature, the gas molecules a velocity component in the opposite direction to our motion appear to increase in velocity relative to us, but the gas molecules with a velocity component in the same direction as we are moving are slower from our point of view and the two effects cancel each other out. The average incoherent kinetic energies of the gas molecules relative to the centre of mass of the gas is unchanged, but there is coherent motion superimposed on top of that, that does not contribute to the temperature.
