  Thermal fluctuations and melting transitions for rotating single-component
superfluids have been intensively studied and are well understood. In contrast,
the thermal effects on vortex states for two-component superfluids with
density-density interaction, which have a much richer variety of vortex ground
states, have been much less studied. Here, we investigate the thermal effects
on vortex matter in superfluids with $\mathrm{U(1)}\times \mathrm{U(1)}$ broken
symmetries and intercomponent density-density interactions, as well as the case
with a larger $\mathrm{SU(2)}$ broken symmetry obtainable from the
$\mathrm{U(1)}\times \mathrm{U(1)}$-symmetric case by tuning scattering
lengths. In the former case we find that, in addition to first-order melting
transitions, the system exhibits thermally driven phase transitions between
square and hexagonal lattices. Our main result, however, concerns the case
where the condensate exhibits $\mathrm{SU(2)}$-symmetry, and where vortices are
not topological. At finite temperature, the system exhibits effects which do
not have a counter-part in single component systems. Namely, it has a state
where thermally averaged quantities show no regular vortex lattice, yet the
system retains superfluid coherence along the axis of rotation. In such a
state, the thermal fluctuations result in transitions between different
(nearly)-degenerate vortex states without undergoing a melting transition. Our
results apply to multi-component Bose-Einstein condensates, and we suggest how
to experimentally detect some of these unusual effects in such systems.
