  In this paper, we investigate the thermodynamics of higher-dimensional $f(R)$
black holes in the extended phase space. Both the analytic expressions and
numerical results for the possible critical physical quantities are obtained.
It is proved that meaningful critical specific volume only exists when $p$ is
odd. This unique phenomenon may be attributed to the combined effect of $f(R)$
gravity and conformally invariant Maxwell field. It is also shown that the
ratio $P_cv_c/T_c$ differs from that of higher dimensional charged AdS black
holes in Einstein gravity. However, the ratio for four-dimensional $f(R)$ black
holes is the same as that of four-dimensional RN-AdS black holes, implying that
$f(R)$ gravity does not influence the ratio. So the ratio may be related to
conformally invariant Maxwell field. To probe the phase transition, we derive
the explicit expression of the Gibbs free energy with its graph plotted. Phase
transition analogous to the van der Waals liquid-gas system take place between
the small black hole and the large black hole. Classical swallow tail behavior,
characteristic of first order phase transition, can also be observed in the
Gibbs free energy graph. Critical exponents are also calculated. It is shown
that these exponents are exactly the same as those of other AdS black holes,
implying that neither $f(R)$ gravity nor conformally invariant Maxwell field
influence the critical exponents. Since the investigated black hole solution
depends on the form of the function $f(R)$, we discuss in detail how our
results put constraint on the form of the function $f(R)$ and we also present a
simple example.
