Topological phononic crystals exhibit remarkable capabilities for manipulating elastic and acoustic waves. In higher-order topological phononic crystals, topological interface and corner states enable wave localization and propagation that are robust against geometric defects and disorders. However, owing to the high-dimensional design space and the complexity of topological properties in two-dimensional phononic crystals, the direct prediction and customized design of their band gap boundaries and topological state frequencies remain extremely challenging. In this work, deep learning approaches are used for the design of two-dimensional phononic crystals with high design freedoms. The considered two-dimensional phononic crystals are composed of periodic solid scatterers embedded in the air background, whose unit cell is divided into a matrix with 32 × 32 pixels. First, the variational autoencoder is applied to reduce the dimensionality of unit cell images, allowing accurate reconstruction of images with different numbers of scatterers. Subsequently, the multilayer perceptron and the tandem neural network are used to realize the property prediction and customized design of two-dimensional phononic crystals, respectively. The correlation coefficients for the property prediction and inverse design are higher than 97%. The unit cell images of two-dimensional phononic crystals with specific band gap properties could be successfully and instantaneously designed. Furthermore, phononic crystals with prescribed topological interface state and corner state frequencies are realized. Experimental measurements are conducted for phononic crystals designed with specific target frequencies, confirming the existence of the designed topological interface and corner states. This study demonstrates the broad application prospects of deep learning approaches in the field of phononic crystal design and provides new ideas and methods for the intelligent design of artificially functional materials.