Thermodynamics of accelerated fermion gas and instability at Unruh temperature

We demonstrate that the energy density of an accelerated fermion gas evaluated within quantum statistical approach in Minkowski space is related to a quantum correction to the vacuum expectation value of the energy-momentum tensor in a space with non-trivial metric and conical singularity. The key element of the derivation is the existence of a novel class of polynomial Sommerfeld integrals. The emerging duality of quantum statistical and geometrical approaches is explicitly checked at temperatures $T$ above or equal to the Unruh temperature $T_U$. Treating the acceleration as an imaginary part of the chemical potential allows for an analytical continuation to temperatures $T<T_U$ . There is a discontinuity at $T=T_U$ manifested in the second derivative of the energy density with respect to the temperature. Moreover, energy density becomes negative at $T<T_U$, apparently indicating some instability. Obtained results might have phenomenological implications for the physics of heavy-ion collisions.