  We obtain a charged, rotating and accelerating black hole solution in the
$f(R)$ gravity and calculate thermodynamic quantities of the black hole in the
slow acceleration regime. We then find that the black hole can be
super-entropic in a certain condition. After investigating the null
hypersurface of the black hole, we show that there exist super-entropic black
holes whose null hypersurface caustics only form inside the Cauchy horizon.
