As a kind of graph structure with timestamps when nodes interact with each other, temporal graphs have more modeling advantages than static graphs. For example, they can detect money laundering, order brushing, equity relationships, financial fraud, and circular guarantees within a certain time interval. The cycle is the modeling of the behavior that forms a cycle in a temporal graph. Existing temporal cycle detection or mining methods mostly focus on the detection of non-decreasing complete cycles in time, but overlook the analysis and discovery of approximate cycles within a certain time interval. The discovery of such approximate cycles can detect fraudulent behavior with stronger cheating techniques. To address the problem of discovering approximate cycles that have already appeared within a certain time interval but are not fully displayed in a single source of data, this study first proposes an approximate cycle detection method based on the depth-first search, which is referred to as the baseline method (Baseline). It first mines complete cycles composed of edges satisfying non-decreasing order in each window, and then employs nodes that meet certain criteria as the start and end points of approximate cycles. In the subsequent windows, paths composed of edges within a certain time interval are mined, namely time-interval approximate cycles. To address the problems of Baseline, this study subsequently proposes an improved method for approximate cycle detection, referred to as the improved method (Improved). It first utilizes the node activity to enhance the possibility of start and end points, then improves the index features by adopting active paths and hotspots, and finally accelerates detection by employing the bidirectional search and connection from start and end points to hotspots. Extensive experiments on real and synthetic data demonstrate the efficiency and effectiveness of the proposed method.