Fractional dark energy: phantom behavior and negative absolute temperature

The fractional dark energy (FDE) model describes the accelerated expansion of the Universe through a nonrelativistic gas of particles with a noncanonical kinetic term. This term is proportional to the absolute value of the three-momentum to the power of $3w$, where $w$ is simply the dark energy equation of state parameter, and the corresponding energy leads to an energy density that mimics the cosmological constant. In this paper we expand the fractional dark energy model considering a non-zero chemical potential and we show that it may thermodynamically describe a phantom regime. The Planck constraints on the equation of state parameter put upper limits on the allowed value of the ratio of the chemical potential to the temperature. In the second part, we investigate the system of fractional dark energy particles with negative absolute temperatures (NAT). NAT are possible in quantum systems and in cosmology, if there exists an upper bound on the energy. This maximum energy is one ingredient of the FDE model and indicates a connection between FDE and NAT, if FDE is composed of fermions. In this scenario, the equation of state parameter is equal to minus one and, using cosmological observations, we find that the transition from positive to negative temperatures is allowed at any redshift larger than one.