Aerial robots are increasingly being utilized for environmental monitoring and exploration. However, a key challenge is efficiently planning paths to maximize the information value of acquired data as an initially unknown environment is explored. To address this, we propose a new approach for informative path planning based on deep reinforcement learning (RL). Combining recent advances in RL and robotic applications, our method combines tree search with an offline-learned neural network predicting informative sensing actions. We introduce several components making our approach applicable for robotic tasks with high-dimensional state and large action spaces. By deploying the trained network during a mission, our method enables sample-efficient online replanning on platforms with limited computational resources. Simulations show that our approach performs on par with existing methods while reducing runtime by 8-10x. We validate its performance using real-world surface temperature data.