We generalize the asymptotic shape theorem in first passage percolation on ℤ^d to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for ℤ^d-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman’s subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon–Nikodym property.