  We investigate the phase diagram of the Haldane-Falicov-Kimball model -- a
model combining topology, interactions and spontaneous disorder at finite
temperatures. Using an unbiased numerical method, we map out the phase diagram
on the interaction--temperature plane. Along with known phases, we unveil an
\textit{insulating charge ordered state with gapless excitations }and a
temperature-driven \textit{gapless topological insulating} phase. Intrinsic --
temperature generated -- disorder, is the key ingredient explaining the
unexpected behavior. Our findings support the possibility of having
temperature-driven topological phase transitions into gapped and gapless
topological insulating phases in systems with a large mass unbalance in
fermionic species.
