  We predict a non-monotonous temperature dependence of the persistent currents
in a ballistic ring coupled strongly to a stub in the grand canonical as well
as in the canonical case. We also show that such a non-monotonous temperature
dependence can naturally lead to a $\phi_0/2$ periodicity of the persistent
currents, where $\phi_0$=h/e. There is a crossover temperature $T^*$, below
which persistent currents increase in amplitude with temperature while they
decrease above this temperature. This is in contrast to persistent currents in
rings being monotonously affected by temperature. $T^*$ is parameter-dependent
but of the order of $\Delta_u/\pi^2k_B$, where $\Delta_u$ is the level spacing
of the isolated ring. For the grand-canonical case $T^*$ is half of that for
the canonical case.
