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This paper considers the problem of cost-aware downlink sum-rate maximization via joint optimal radio access technologies (RATs) assignment and power allocation in next-generation heterogeneous wireless networks (HetNets). We consider a future HetNet comprised of multi-RATs and serving multi-connectivity edge devices (EDs), and we formulate the problem as mixed-integer non-linear programming (MINP) problem. Due to the high complexity and combinatorial nature of this problem and the difficulty to solve it using conventional methods, we propose a hierarchical multi-agent deep reinforcement learning (DRL)-based framework, called DeepRAT, to solve it efficiently and learn system dynamics. In particular, the DeepRAT framework decomposes the problem into two main stages; the RATs-EDs assignment stage, which implements a single-agent Deep Q Network (DQN) algorithm, and the power allocation stage, which utilizes a multi-agent Deep Deterministic Policy Gradient (DDPG) algorithm. Using simulations, we demonstrate how the various DRL agents efficiently interact to learn system dynamics and derive the global optimal policy. Furthermore, our simulation results show that the proposed DeepRAT algorithm outperforms existing state-of-the-art heuristic approaches in terms of network utility. Finally, we quantitatively show the ability of the DeepRAT model to quickly and dynamically adapt to abrupt changes in network dynamics, such as EDs mobility.