Implementations of artificial intelligence (AI) based on deep learning (DL) have proven to be highly successful in many domains, from biomedical imaging to natural language processing, but are still rarely applied in the space industry, particularly for onboard learning of planetary surfaces. In this project, we discuss the utility and limitations of DL, enhanced with topological footprints of the sensed objects, for multi-class classification of planetary surface patterns, in conjunction with tactile and embedded sensing in rover exploratory missions. We consider a Topological Convolutional Network (TCN) model with a persistence-based attention mechanism for supervised classification of various landforms. We study TCN's performance on the Barefoot surface pattern dataset, a novel surface pressure dataset from a prototype tactile rover wheel, known as the Barefoot Rover tactile wheel. Multi-class pattern recognition in the Barefoot data has neither been ever tackled before with DL nor assessed with topological methods. We provide insights into advantages and restrictions of topological DL as the early-stage concept for onboard learning and planetary exploration.