Building on parallels between geometric quantum mechanics and classical mechanics, we explore an alternative basis for quantum thermodynamics that exploits the differential geometry of the underlying state space. We focus on microcanonical and canonical ensembles, looking at the geometric counterpart of Gibbs ensembles for distributions on the space of quantum states. We show that one can define quantum heat and work in an intrinsic way, including single-trajectory work. We reformulate thermodynamic entropy in a way that accords with classical, quantum, and information-theoretic entropies. We give both the first and second laws of thermodynamics and Jarzynki's fluctuation theorem. Overall, this results in a more transparent physics than conventionally available. The mathematical structure and physical intuitions underlying classical and quantum dynamics are seen to be closely aligned. The experimental relevance is brought out via a stochastic model for chiral molecules (in the two-state approximation) and Josephson junctions. Numerically, we demonstrate this invariably leads to the emergence of the geometric canonical ensemble.