With massive smart devices accessing the Internet, multi-tier computing has shown promise as a system-level network architecture by deploying caching capabilities at edge servers that are closer to the user. However, constrained by factors such as the dynamic nature of user content requests and limited cache resources, cache allocation is key to improving network performance. To address these challenges, we investigate and formulate the cache allocation problem for maximizing the cache hit ratio in mobile edge-cloud computing networks. We propose a framework to address this issue based on double deep Q-networks. The framework adaptively allocates cache slots to cloud content providers (CCPs) based on variations in user demand for different CCPs and temporal fluctuations in this demand. Simulation results depict a measurable enhancement in the cache hit ratio of the proposed framework compared to the baseline frameworks.