  We study the evolution of the doubly charmed state $T_{cc}^+$ in a hot hadron
gas produced in the late stage of heavy-ion collisions. We use effective
Lagrangians to calculate the thermally averaged cross sections of $T_{cc}^+$
production in reactions such as $ D^{(*)} D^{(*)} \rightarrow T_{cc}^+ \pi,
T_{cc}^+ \rho $ and its absorption in the corresponding inverse processes. We
then solve the rate equation to follow the time evolution of the $T_{cc}^+$
multiplicity, and determine how it is affected by the considered reactions
during the expansion of the hadronic matter. We compare the evolution of the
$T_{cc}^+$ abundance treated as a hadronic $S$-wave molecule and as a
tetraquark state. Our results show that the tetraquark yield increases by a
factor of about 2 at freeze-out, but it is still one order of magnitude smaller
than the final yield of molecules formed from hadron coalescence. Also, the
results indicate that $N_{T_{cc}}$ is more affected by interactions with
hadronic medium than $ N_{X(3872)} $ in similar conditions. We also show that
yields depend very weakly on the system size, represented by $\mathcal{N} =
\left[ d N_{ch} / d \eta (\eta < 0.5)\right]^{1/3}$.
