I was reading: https://plato.stanford.edu/entries/equivME/#PhysMassEnerEqui
Where they begin to discuss a weighing a relativistic gas against a cold gas.
The idea is that the relativistic gas molecules are moving faster and therefore their "relativistic masses" are higher and therefore they weigh more, and that this could quite literally be measured by considering a balance with the same # of molecules and same container holding them, only one container is at a high temperature and the other at a low temperature.
Why I think this is suspect:
At best if you bring out full on General Relativity then the internal pressure of the gas might have some non trivial gravitational effect to make it heavier, but I DO NOT believe that the hotter gas will be heavier on just purely Special Relativity + Newtonian Grounds. ORIGINAL ARGUMENT REDACTED BECAUSE I FOUND A FLAW.
Original Text Below:
"Assuming the principle of equivalence, which entails that we can measure inertial mass by measuring gravitational mass with a balance, we can illustrate the difference between the Newtonian and relativistic understanding of the ideal gas as follows. Imagine that we have two otherwise identical massless vessels filled with exactly the same amount and type of gas. In one vessel, the gas is at a temperature very near absolute zero, so its molecules have very little kinetic energy. In the other vessel, the gas is at a temperature of 500° C. Place these two vessels of gas on the ends of a balance. According to Newtonian physics, the balance will be level, because both gas samples have exactly the same mass. According to relativity, the balance will not be level and will be tipped on the side of the hot gas, because the high kinetic energy of the molecules contributes to the rest energy of the gas, which contributes, through Einstein’s equation, to the rest-mass of the vessel of gas."