In multi-domain learning (MDL) scenarios, high labeling effort is required due to the complexity of collecting data from various domains. Active Learning (AL) presents an encouraging solution to this issue by annotating a smaller number of highly informative instances, thereby reducing the labeling effort. Previous research has relied on conventional AL strategies for MDL scenarios, which underutilize the domain-shared information of each instance during the selection procedure. To mitigate this issue, we propose a novel perturbation-based two-stage multi-domain active learning (P2S-MDAL) method incorporated into the well-regarded ASP-MTL model. Specifically, P2S-MDAL involves allocating budgets for domains and establishing regions for diversity selection, which are further used to select the most cross-domain influential samples in each region. A perturbation metric has been introduced to evaluate the robustness of the shared feature extractor of the model, facilitating the identification of potentially cross-domain influential samples. Experiments are conducted on three real-world datasets, encompassing both texts and images. The superior performance over conventional AL strategies shows the effectiveness of the proposed strategy. Additionally, an ablation study has been carried out to demonstrate the validity of each component. Finally, we outline several intriguing potential directions for future MDAL research, thus catalyzing the field's advancement.