Anomaly detection in dynamic networks is critical for applications from cybersecurity to industrial monitoring, yet existing methods face challenges in energy efficiency, temporal precision, and adaptability. This paper introduces ASTDP-GAD, a novel Adaptive Spiking Temporal Dynamics Plasticity framework for Graph Anomaly Detection that integrates spiking graph neural networks with STDP learning for energy-efficient neuromorphic detection in dynamic networks. Our framework unifies spiking neural computation, STDP learning, and graph-based anomaly detection through the following key innovations: temporal spike graph encoding with adaptive Leaky Integrate-and-Fire (LIF) dynamics; LIF-based graph attention with lateral inhibition; event-driven hypergraph memory with STDP-inspired prototype updates; spike rate contrast pooling based on spiking irregularity; adaptive STDP layers capturing causal temporal relationships; and multi-scale temporal convolution with multi-factor anomaly fusion. Theoretical analysis provides rigorous guarantees: spike encoding preserves input information with resolution scaling linearly in simulation steps and hidden dimension; LIFGAT approximates any continuous attention function; hypergraph memory converges to optimal prototypes; contrast pooling achieves provable anomaly selection bounds; STDP learning converges stably; and multi-factor fusion produces calibrated scores with up to $5\times$ variance reduction. Extensive experiments on nine datasets on both dynamic and static graphs demonstrate superior anomaly detection accuracy while maintaining biological plausibility and energy efficiency for neuromorphic deployment.