This paper addresses the task of classifying galaxy clusters, which are the largest known objects in the Universe. Galaxy Clusters can be categorized into cool-core (CC), weak-cool-core (WCC), and non-cool-core (NCC) clusters, defined by their central cooling times. Conventional approaches in astrophysics for conducting such classification are through measuring their surface brightness concentrations or central gas densities, which are inaccurate. Off-the-shelf deep learning approaches for solving this problem would be taking entire images as inputs and predicting cluster types directly. However, this strategy is limited in that central cooling times are usually related to only small informative regions near the center, and feeding unrelated outer regions into the network may hurt the performance. In this work, we propose multi-branch attention networks that utilize attention and bivariate Gaussian distribution to identify the galaxy cluster type. Our loss function is designed by encompassing our domain knowledge that the central cooling time of three different types of galaxy clusters (CC, WCC, NCC) varies continuously. To train our model, we take advantage of simulated X-ray emissivity images, which contains 954 projections of 318 clusters. Experimental results demonstrate that our method outperforms the baseline method and can be applied to classify large galaxy images accurately.