  We find a finite free resolution of the counit of the free unitary quantum
groups of van Daele and Wang and, more generally, Bichon's universal
cosovereign Hopf algebras with a generic parameter matrix. This allows us to
compute Hochschild cohomology with 1-dimensional coefficients for all these
Hopf algebras. In fact, the resolutions can be endowed with a Yetter-Drinfeld
structure. General results of Bichon then allow us to compute also the
corresponding bialgebra cohomologies.
  Finding the resolution rests on two pillars. We take as a starting point the
resolution for the free orthogonal quantum group presented by Collins,
H\"artel, and Thom or its algebraic generalization to quantum symmetry groups
of bilinear forms due to Bichon. Then we make use of the fact that the free
unitary quantum groups and some of its non-Kac versions can be realized as a
glued free product of a (non-Kac) free orthogonal quantum group with $\mathbb
Z_2$, the finite group of order 2. To obtain the resolution also for more
general universal cosovereign Hopf algebras, we extend Gromada's proof from
compact quantum groups to the framework of matrix Hopf algebras. As a byproduct
of this approach, we also obtain a projective resolution for the freely
modified bistochastic quantum groups.
  Only a special subclass of free unitary quantum groups and universal
cosovereign Hopf algebras decompose as a glued free product in the described
way. In order to verify that the sequence we found is a free resolution in
general (as long as the parameter matrix is generic, two conditions which are
automatically fulfilled in the free unitary quantum group case), we use the
theory of Hopf bi-Galois objects and Bichon's results on monoidal equivalences
between the categories of Yetter-Drinfeld modules over universal cosovereign
Hopf algebras for different parameter matrices.
