Open-vocabulary semantic segmentation presents the challenge of labeling each pixels within an image based on wide range of text descriptions. In this work, we introduce a novel cost-based approach to adapt vision-language foundation models, notably CLIP, for the intricate task of semantic segmentation. Through aggregating the cosine similarity score, i.e. the cost volume between image and text embeddings, our method potently adapts CLIP for segmenting seen and unseen classes by fine-tuning its encoders, addressing the challenges faced by existing methods in handling unseen classes. Building upon this, we explore methods to effectively aggregate the cost volume considering its multi-modal nature of being established between image and text embeddings. Furthermore, we examine various methods for efficiently fine-tuning CLIP. Our framework, dubbed CAT-Seg, shows state-of-the-art performance on standard benchmarks with significant margins, and further exerts strengths in more challenging scenarios from various domains.