Continual test-time adaptive object detection (CTTA-OD) aims to online adapt a source pre-trained detector to ever-changing environments during inference under continuous domain shifts. Most existing CTTA-OD methods prioritize effectiveness while overlooking computational efficiency, which is crucial for resource-constrained scenarios. In this paper, we propose an efficient CTTA-OD method via pruning. Our motivation stems from the observation that not all learned source features are beneficial; certain domain-sensitive feature channels can adversely affect target domain performance. Inspired by this, we introduce a sensitivity-guided channel pruning strategy that quantifies each channel based on its sensitivity to domain discrepancies at both image and instance levels. We apply weighted sparsity regularization to selectively suppress and prune these sensitive channels, focusing adaptation efforts on invariant ones. Additionally, we introduce a stochastic channel reactivation mechanism to restore pruned channels, enabling recovery of potentially useful features and mitigating the risks of early pruning. Extensive experiments on three benchmarks show that our method achieves superior adaptation performance while reducing computational overhead by 12% in FLOPs compared to the recent SOTA method.