  Higher-order exceptional points (EPs), resulting from non-Hermitian
degeneracies, have shown greater advantages in sensitive enhancement than
second-order EPs (EP2s). Therefore, seeking higher-order EPs in various quantum
systems is important for quantum information science. Here we propose a
benchmark cavity optomechanical (COM) system consisting of a mechanical
resonator (MR) coupled to two cavities via radiation pressure for predicting
the third-order exceptional point (EP3). We first give the pseudo-Hermitian
condition for the non-Hermitian COM system by taking the bath effects into
account. Then we consider the mechanical gain effect and we find that the
pseudo-Hermitian COM system without $\mathcal{PT}$ symmetry can host both the
EP3 and EP2 for symmetric and asymmetric cavities. In the symmetric case, only
the EP3 or EP2 can be predicted in the parameter space, but the EP3 and EP2 can
be transformed into each other by tuning the COM coupling strength in the
asymmetric case. We further consider the case of one cavity with gain. For this
case, the pseudo-Hermitian COM system is $\mathcal{PT}$-symmetric and can also
host the EP3 or EP2. The influence of system parameters on them are discussed.
Our proposal provides a potential way to realize sensitive detection and study
other physical phenomena {around} higher-order EP3 in non-Hermitian COM
systems.
