Since 1940, the Kolmogorov's statistical formalism of exact law that describes incompressible hydrodynamic turbulence through the energy cascade rate, has been extended to compressible magnetized fluid described by isothermal or polytropic closure. Its estimation in the solar wind helped to better understand particle heating in such collisionless media assuming that all the energy is eventually dissipated at the smallest scales by some (kinetic) effects. However, considering an isotropic pressure (with an isothermal or polytropic closure) can be questioned in space plasmas where pressure anisotropy is frequently observed.

We propose a general exact law of Hall-MHD turbulence based on models with a pressure tensor that allows us to study various known equations of state as particular limits, derive a new one corresponding to the CGL closure (i.e., gyrotropic pressure tensor), a model that potentially paves the road to linking the turbulent cascade to plasma instabilities driven by pressure anisotropy. In the incompressible MHD limit we provide a generalization of the so-called Politano & Pouquet law to pressure-anisotropic plasmas.