We rigorously compute the integrable system for the limiting(N\→\∞) distribution function of the extreme momentum of N fermions when confined to an anharmonic trap V(q)=q²ⁿ for\∈\≥ 1 at positive temperature. More precisely, the edge statistics in the harmonic trap $n=1$ are known to obey the weak KPZ crossover law which is realized via the finite temperature Airy determinant or equivalently via a Painlev\\'e-II integro-differential, cf. \{LW,ACQ}. For general n\≥ 2, a novel higher order temperature Airy kernel has recently emerged in physics literature\{DMS} and we show that the corresponding edge law in momentum space is now by a distinguished Painlev\\'e-II integro-differential hierarchy. Our is based on operator-valued Riemann-Hilbert techniques which produce a pair for an operator-valued Painlev\\'e-II ODE system that naturally encodes aforementioned hierarchy. As byproduct, we establish a connection of the-differential Painlev\\'e-II hierarchy to a novel integro-differential hierarchy.