  The solutions of the Einstein equation are a subset of the solutions of
conformal (Weyl) gravity, but the difference from the action means that the
black hole thermodynamics of the two gravity theories would be different. In
this paper we explore the thermodynamic phase structure for the conformal
gravity in the four-dimensional AdS space-time. Special emphasis is put on the
dependence on the parameter $c_1$ in the linear-$r$ term in the metric. The
thermodynamic phase structure of the conformal gravity is very rich, including
two branches of equations of states, negative thermodynamic volume,
zeroth-order phase transition, and Hawking-Page like phase transition.
