Scaling laws for language models have often focused on finding the optimal model size and token count for training from scratch. However, achieving this optimal balance requires significant compute resources due to the extensive data demands when training models from randomly-initialized weights. Continued pretraining offers a cost-effective alternative, leveraging the compute investment from pretrained models to incorporate new knowledge without requiring extensive new data. Recent findings suggest that data quality influences constants in scaling laws, thereby altering the optimal parameter-token allocation ratio. Building on this insight, we investigate the interplay between domain specialization and model size during continued pretraining under compute-constrained scenarios. Our goal is to identify an optimal training regime for this scenario and detect patterns in this interplay that can be generalized across different model sizes and domains. To compare general and specialized training, we filtered a web-based dataset to extract data from three domains: legal, medical, and accounting. We pretrained models with 1.5B, 3B, 7B, and 14B parameters on both the unfiltered and filtered datasets, then evaluated their performance on domain-specific exams. Results show that as model size increases, specialized models outperform general models while requiring less training compute. Additionally, their growing compute efficiency leads to reduced forgetting of previously learned knowledge.