Yeah it’s a weird one, made me think as well and I’m definitely not 100% sure about this but basically the the keq only changes if the temperature changes, which would only change if someone added an EXTERNAL change in temperature. And if the concentrations of everything is still the same then the keq is still the same so the temperature has to be the same.
Think of it in terms of collision theory - sure more particles are bumping into each other, more of those are actually with the inert gas which it won’t react with, so in the end there is no net effect.
Remember PV=nRT? Because n increases, pressure increases, so it stays balanced WITHOUT a change in temperature. No change in temperature and no change in concentrations means no change in Keq