Photon-counting computed tomography (PCCT) reconstructs multiple energy-channel images to describe the same object, where there exists a strong correlation among different channel images. In addition, reconstruction of each channel image suffers photon count starving problem. To make full use of the correlation among different channel images to suppress the data noise and enhance the texture details in reconstructing each channel image, this paper proposes a tensor neural network (TNN) architecture to learn a multi-channel texture prior for PCCT reconstruction. Specifically, we first learn a spatial texture prior in each individual channel image by modeling the relationship between the center pixels and its corresponding neighbor pixels using a neural network. Then, we merge the single channel spatial texture prior into multi-channel neural network to learn the spectral local correlation information among different channel images. Since our proposed TNN is trained on a series of unpaired small spatial-spectral cubes which are extracted from one single reference multi-channel image, the local correlation in the spatial-spectral cubes is considered by TNN. To boost the TNN performance, a low-rank representation is also employed to consider the global correlation among different channel images. Finally, we integrate the learned TNN and the low-rank representation as priors into Bayesian reconstruction framework. To evaluate the performance of the proposed method, four references are considered. One is simulated images from ultra-high-resolution CT. One is spectral images from dual-energy CT. The other two are animal tissue and preclinical mouse images from a custom-made PCCT systems. Our TNN prior Bayesian reconstruction demonstrated better performance than other state-of-the-art competing algorithms, in terms of not only preserving texture feature but also suppressing image noise in each channel image.