Wouldn't gamma be T dependent, as more modes become 'available'.
Sure. The whole idea of "polytropic" is just an
approximation, valid over a _limited_ temperature
range ... but sometimes the range is large enough
have genuine practical value. In particular, it's
not a coincidence that we have gamma = 1.4 for N2,
O2, CO, and dry air under a wide range of ordinary
conditions.