  We give a universal description of the mesoscopic effects occurring in
fractional quantum Hall disks due to the Aharonov-Bohm flux threading the
system. The analysis is based on the exact treatment of the flux within the
conformal field theory framework and is relevant for all fractional quantum
Hall states whose edge states CFTs are known. As an example we apply this
scheme for the parafermion Hall states and extract the main characteristics of
the low- and high- temperature asymptotic behavior of the persistent currents.
