  We derive a first exact law for compressible pressure-anisotropic
magnetohydrodynamic turbulence. For a gyrotropic pressure tensor, we study the
double-adiabatic case and show the presence of new flux and source terms in the
exact law, reminiscent of the plasma instability conditions due to pressure
anisotropy. The Hall term is shown to bring ion-scale corrections to the exact
law without affecting explicitly the pressure terms. In the pressure isotropy
limit we recover all known results obtained for isothermal and polytropic
closures. The incompressible limit of the gyrotropic system leads to a
generalization of the Politano and Pouquet's law where a new {\it
incompressible} source term is revealed and reflects exchanges of the magnetic
and kinetic energies with the no-longer-conserved internal energy. We highlight
the possibilities offered by the new laws to investigate potential links
between turbulence cascade and instabilities widely observed in laboratory and
astrophysical plasmas.
