Explainable artificial intelligence (XAI) enhances AI system transparency by framing interpretability as an optimization problem. However, this approach often necessitates numerous iterations of computationally intensive operations, limiting its applicability in real-time scenarios. While recent research has focused on XAI hardware acceleration on FPGAs and TPU, these methods do not fully address energy efficiency in real-time settings. To address this limitation, we propose XAIedge, a novel framework that leverages approximate computing techniques into XAI algorithms, including integrated gradients, model distillation, and Shapley analysis. XAIedge translates these algorithms into approximate matrix computations and exploits the synergy between convolution, Fourier transform, and approximate computing paradigms. This approach enables efficient hardware acceleration on TPU-based edge devices, facilitating faster real-time outcome interpretations. Our comprehensive evaluation demonstrates that XAIedge achieves a $2\times$ improvement in energy efficiency compared to existing accurate XAI hardware acceleration techniques while maintaining comparable accuracy. These results highlight the potential of XAIedge to significantly advance the deployment of explainable AI in energy-constrained real-time applications.