This paper investigates the impact of hand gestures and eye gazing on cognitive load and task efficiency in Augmented Reality (AR) using Microsoft’s HoloLens 2 and the custom-developed application MazeMind. By conducting human-subject experiments in MazeMind, we assessed cognitive load using NASA-TLX and Galvanic Skin Response (GSR), confirming a significant correlation between real-time GSR readings and post-interaction NASA-TLX scores. This underscores GSR's potential for real-time cognitive load assessment in AR. Contrary to the expectations, our findings indicated that there was no significant difference between hand-gesture-based and eye-gaze-based interactions in terms of cognitive load and completion efficiency in AR activities, even for cognitively challenging tasks. This suggests the potential interchangeability of these interaction modalities in AR and provides evidence-based guidelines for designers. Our study contributes to a deeper understanding of multimodal interactions in AR and lays the foundation for future exploration of more complex AR interaction design strategies.