  We consider the effect of thermal fluctuations on rotating spinor F=1
condensates in axially-symmetric vortex phases, when all the three hyperfine
states are populated. We show that the relative phase among different
components of the order parameter can fluctuate strongly due to the weakness of
the interaction in the spin channel. These fluctuations can be significant even
at low temperatures. Fluctuations of relative phase lead to significant
fluctuations of the local transverse magnetization of the condensate. We
demonstrate that these fluctuations are much more pronounced for the
antiferromagnetic state than for the ferromagnetic one.
