  We study mechanical cooling in systems of coupled passive (lossy) and active
(with gain) optical resonators. We find that for a driving laser which is
red-detuned with respect to the cavity frequency, the supermode structure of
the system is radically changed, featuring the emergence of genuine high-order
exceptional points. This in turn leads to giant enhancement of both the
mechanical damping and the spring stiffness, facilitating low-power mechanical
cooling in the vicinity of gain-loss balance. This opens up new avenues of
steering micromechanical devices with exceptional points beyond the
lowest-order two.
