  We generalize the asymptotic shape theorem in first passage percolation on
$\mathbb{Z}^d$ 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 $\mathbb{Z}^d$-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.
