This paper shows the importance of considering the thermal state of a combustor to investigate or predict its thermoacoustic stability. This aspect is often neglected or regarded as less important than the effect of the operating parameters, such as thermal power or equivalence ratio, but under certain circumstances it can have a dramatic influence on the development of the instabilities. The paper presents experimental results collected from a combustor featuring a lean swirl-stabilized flame exhibiting thermoacoustic instability at some operating conditions. It is shown that this instability is caused by a change of the flame topology that is induced by the progressive increase of the wall temperature with the thermal power. This dependence of the instability on wall temperature leads to inertial effects and hysteresis when the operating condition is changed dynamically. A low-order model of the system reproducing this remarkable dynamics is proposed and validated against the experimental data.