Objective: Today, stress monitoring on wearable devices is challenged by the tension between high-detection accuracy and battery lifetime driven by multimodal data acquisition and processing. Limited research has addressed the classification cost on multimodal wearable sensors, particularly when the features are cost-dependent. Thus, we design a Cost-Aware Feature Selection (CAFS) methodology that trades-off between prediction-power and energy-cost for multimodal stress monitoring. Methods: CAFS selects the most important features under different energy-constraints, which allows us to obtain energy-scalable stress monitoring models. We further propose a self-aware stress monitoring method that intelligently switches among the energy-scalable models, reducing energy consumption. Results: Using CAFS methodology on experimental data and simulation, we reduce the energy-cost of the stress model designed without energy constraints up to 94.37%. We obtain 90.98% and 95.74% as the best accuracy and confidence values, respectively , on unseen data, outperforming state-of-the-art studies. Analyzing our interpretable and energy-scalable models, we showed that simple models using only heart rate (HR) or skin conductance level (SCL), confidently predict acute stress for HR > 93.30BP M and non-stress for SCL < 6.42µS, but, outside these values, a multimodal model using respiration and pulse wave's features is needed for confident classification. Our self-aware acute stress monitoring proposal saves 10x energy and provides 88.72% of accuracy on unseen data. Conclusion: We propose a comprehensive solution for the cost-aware acute stress monitoring design addressing the problem of selecting an optimized feature subset considering their cost-dependency and cost-constraints. Significant: Our design framework enables long-term and confident acute stress monitoring on wearable devices.