Infrared and visible video fusion plays a critical role in intelligent surveillance and low-light monitoring. However, maintaining temporal stability while preserving spatial detail remains a fundamental challenge. Existing methods either focus on frame-wise enhancement with limited temporal modeling or rely on heavy spatio-temporal aggregation that often sacrifices high-frequency details. In this paper, we propose FTPFusion, a frequency-aware infrared and visible video fusion method based on temporal perturbation and sparse cross-modal interaction. Specifically, FTPFusion decomposes the feature representations into high-frequency and low-frequency components for collaborative modeling. The high-frequency branch performs sparse cross-modal spatio-temporal interaction to capture motion-related context and complementary details. The low-frequency branch introduces a temporal perturbation strategy to enhance robustness against complex video variations, such as flickering, jitter, and local misalignment. Furthermore, we design an offset-aware temporal consistency constraint to explicitly stabilize cross-frame representations under temporal disturbances. Extensive experiments on multiple public benchmarks demonstrate that FTPFusion consistently outperforms state-of-the-art methods across multiple metrics in both spatial fidelity and temporal consistency. The source code will be available at https://github.com/ixilai/FTPFusion.