  We investigate the adiabatic evolution of thermal state in non-reciprocal
many-body systems coupled to their environment and subject to periodic
drivings. In such systems we show that besides the dynamical phase a
geometrical phase can exist and it drives the relaxation dynamic of the system.
On the contrary to the dynamical phase which always pushes the system toward
its equilibrium state we show that the geometric phase can speed up or reduce
the speed of relaxation process. These results could have applications in the
field of thermal management of complex systems.
