The Amazon rainforest is considered one of the Earth's tipping elements and may lose stability under ongoing climate change. Recently a decrease in tropical rainforest resilience has been identified globally from remotely sensed vegetation data. However, the underlying theory assumes a Gaussian distribution of forest disturbances, which is different from most observed forest stressors such as fires, deforestation, or windthrow. Those stressors often occur in power-law-like distributions and can be approximated by α-stable Lévy noise. Here, we show that classical critical slowing down (CSD) indicators to measure changes in forest resilience are robust under such power-law disturbances. To assess the robustness of CSD indicators, we simulate pulse-like perturbations in an adapted and conceptual model of a tropical rainforest. We find few missed early warnings and few false alarms are achievable simultaneously if the following steps are carried out carefully: first, the model must be known to resolve the timescales of the perturbation. Second, perturbations need to be filtered according to their absolute temporal autocorrelation. Third, CSD has to be assessed using the non-parametric Kendall-τ slope. These prerequisites allow for an increase in the sensitivity of early warning signals. Hence, our findings imply improved reliability of the interpretation of empirically estimated rainforest resilience through CSD indicators.