Pinching antenna systems have attracted much attention recently owing to its capability to maintain reliable line-of-sight (LoS) communication in high-frequency bands. By guiding signals through a waveguide and emitting them via a movable pinching antenna, these systems enable dynamic control of signal propagation and spatial adaptability. However, their performance heavily depends on effective resource allocation-encompassing power, bandwidth, and antenna positioning-which becomes challenging under imperfect channel state information (CSI) and user localization uncertainty. Existing studies largely assume perfect CSI or ideal user positioning, while our prior work considered uniform localization errors, an oversimplified assumption. In this paper, we develop a robust resource allocation framework for multiuser downlink pinching antenna systems under Gaussian-distributed localization uncertainty, which more accurately models real-world positioning errors. An energy efficiency (EE) maximization problem is formulated subject to probabilistic outage constraints, and an analytical power allocation strategy is derived under given antenna positions. On this basis, the heuristic particle swarm optimization (PSO) algorithm is employed to identify the antenna position that achieves the global EE configuration. Simulation results illustrate that the proposed scheme greatly enhances both EE and system reliability compared with fixed-antenna benchmark, validating its effectiveness for practical high-frequency wireless deployments.