Multi-phase computed tomography (CT) has been widely used for the preoperative diagnosis of kidney cancer due to its non-invasive nature and ability to characterize renal lesions. However, since enhancement patterns of renal lesions across CT phases are different even for the same lesion type, the visual assessment by radiologists suffers from inter-observer variability in clinical practice. Although deep learning-based approaches have been recently explored for differential diagnosis of kidney cancer, they do not explicitly model the relationships between CT phases in the network design, limiting the diagnostic performance. In this paper, we propose a novel lesion-aware cross-phase attention network (LACPANet) that can effectively capture temporal dependencies of renal lesions across CT phases to accurately classify the lesions into five major pathological subtypes from time-series multi-phase CT images. We introduce a 3D inter-phase lesion-aware attention mechanism to learn effective 3D lesion features that are used to estimate attention weights describing the inter-phase relations of the enhancement patterns. We also present a multi-scale attention scheme to capture and aggregate temporal patterns of lesion features at different spatial scales for further improvement. Extensive experiments on multi-phase CT scans of kidney cancer patients from the collected dataset demonstrate that our LACPANet outperforms state-of-the-art approaches in diagnostic accuracy.