  We study the thermodynamics of spherically symmetric, neutral and
non-rotating black holes in conformal (Weyl) gravity. To this end, we apply
different methods: (i) the evaluation of the specific heat; (ii) the study of
the entropy concavity; (iii) the geometrical approach to thermodynamics known
as \textit{thermodynamic geometry}; (iv) the Poincar\'{e} method that relates
equilibrium and out-of-equilibrium thermodynamics. We show that the
thermodynamic geometry approach can be applied to conformal gravity too,
because all the key thermodynamic variables are insensitive to Weyl scaling.
The first two methods, (i) and (ii), indicate that the entropy of a de Sitter
black hole is always in the interval $2/3\leq S\leq 1$, whereas thermodynamic
geometry suggests that, at $S=1$, there is a second order phase transition to
an Anti de Sitter black hole. On the other hand, we obtain from the
Poincar\'{e} method (iv) that black holes whose entropy is $S < 4/3$ are stable
or in a saddle-point, whereas when $S>4/3$ they are always unstable, hence
there is no definite answer on whether such transition occurs.
