  We show that the effective potentials for the Polyakov loops in finite
temperature SU$(N)$ gauge theories obey a certain scaling relation with respect
to temperature in the large-$N$ limit. This scaling relation strongly
constrains the possible terms in the Polyakov loop effective potentials.
Moreover, by using the effective potentials in the presence of imaginary
chemical potentials or imaginary angular velocities in several models, we find
that phase transitions to $Z_m$-type deconfinement phases ($Z_m$ phase) occur,
where the eigenvalues of the Polyakov loop are distributed $Z_m$ symmetrically.
Physical quantities in the $Z_m$ phase obey the scaling properties of the
effective potential. The models include Yang-Mills (YM) theories, the bosonic
BFSS matrix model and ${\mathcal N}=4$ supersymmetric YM theory on $S^3$. Thus,
the phase diagrams of large-$N$ gauge theories with imaginary chemical
potentials are very rich and the stable $Z_m$ phase would be ubiquitous.
Monte-Carlo calculations also support this. As a related topic, we discuss the
phase diagrams of large-$N$ YM theories with real angular velocities in finite
volume spaces.
