This paper investigates an energy-efficient beamforming design for a hardware-impaired simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-aided integrated sensing and communication (ISAC) system, where the base station (BS) concurrently performs target sensing and multi-user communication. Accounting for hardware impairments (HWIs) at the BS, user equipments (UEs) and STAR-RIS, a joint optimization problem is posed to maximize the system energy efficiency, subject to constraints on required sensing and transmission capabilities, and the total power budget. To tackle the intractable conflicts among sensing and transmission metrics introduced by HWIs, we propose a novel learning-based method that integrates a denoising diffusion probabilistic model (DDPM) into a twin-delayed deep deterministic policy gradient (TD3) algorithm enhanced with prioritized experience replay (PER). By leveraging the DDPM and PER for beamforming policy determination, our approach accurately models the complex dynamics, achieving a better balance between sensing and communication performance. Simulation results demonstrate that the proposed PER-DDPM-TD3-based beamforming strategy achieves a 69.3% higher energy efficiency performance than the existing deep reinforcement learning (DRL)-based method.