Out-Of-Distribution (OOD) detection is critical to deploy deep learning models in safety-critical applications. OOD typically works by minimising intra-class and maximising inter-class separation, which can profit from the exponential volume growth in Hyperbolic space. This work proposes a metric framework that leverages the strengths of Hyperbolic geometry for OOD detection. Inspired by previous works that refine the decision boundary for OOD data with synthetic outliers, we extend this method to Hyperbolic space. Interestingly, we find that synthetic outliers do not benefit OOD detection in Hyperbolic space as they do in Euclidean space. Furthermore we explore the relationship between OOD detection performance and Hyperbolic embedding dimension, addressing practical concerns in resource-constrained environments. Extensive experiments show that our framework improves the FPR95 for OOD detection from 21% to 16% and from 49% to 28% on CIFAR-10 and CIFAR-100 respectively compared to Euclidean methods.