Ecological stability contains multiple components, such as temporal invariability, resistance and resilience. Understanding the response of stability components to perturbations is beneficial for optimizing the management of biodiversity and ecosystem functioning. Although previous studies have investigated the effects of multiple perturbations on each stability component, few studies simultaneously measure the multiple stability components and their relationships. Alpine grasslands on the Tibetan Plateau are exposed to co-occurring perturbations, including climate change and human activities. Here, we quantified three stability components (temporal invariability, resistance, and resilience) of alpine grasslands on the Tibetan Plateau during periods of high (2000-2008) and low (2009-2017) human activity intensity, respectively. We focused on the effects of climate variables (temperature, precipitation, radiation) and human activities (grazing intensity) on covariation among stability components. The results show that (1) for periods of high and low human activity, temporal invariability was positively correlated with resistance and resilience, while resistance was independent of resilience; (2) the dimensionality of alpine grasslands decreased by almost 10%, from 0.61 in the first period to 0.55 in the second period, suggesting the increasing connections among temporal invariability, resistance and resilience of alpine grasslands; and (3) temperature but not grazing intensity dominated the changes in the dimensionality of stability. These findings improve our understanding of multi-dimensional stability and highlight the importance of climate variability on alpine grassland stability on the Tibetan Plateau.