  The theory of multidimensional Poisson vertex algebras (mPVAs) provides a
completely algebraic formalism to study the Hamiltonian structure of PDEs, for
any number of dependent and independent variables. In this paper, we compute
the cohomology of the PVAs associated with two-dimensional, two-components
Poisson brackets of hydrodynamic type at the third differential degree. This
allows us to obtain their corresponding Poisson-Lichnerowicz cohomology, which
is the main building block of the theory of their deformations. Such a
cohomology is trivial neither in the second group, corresponding to the
existence of a class of not equivalent infinitesimal deformation, nor in the
third, corresponding to the obstruction to extend such deformations
