  Given a set of differential forms on an odd-dimensional noncommutative
manifold valued in an internal associative algebra H, we show that the most
general cubic covariant Hamiltonian action, without mass terms, is controlled
by an Z_2-graded associative algebra F with a graded symmetric nondegenerate
bilinear form. The resulting class of models provide a natural generalization
of the Frobenius-Chern-Simons model (FCS) that was proposed in arXiv:1505.04957
as an off-shell formulation of the minimal bosonic four-dimensional higher spin
gravity theory. If F is unital and the Z_2-grading is induced from a Klein
operator that is outer to a proper Frobenius subalgebra, then the action can be
written on a form akin to topological open string field theory in terms of a
superconnection valued in the direct product of H and F. We give a new model of
this type based on a twisting of C[Z_2 x Z_4], which leads to self-dual
complexified gauge fields on AdS_4. If F is 3-graded, the FCS model can be
truncated consistently as to zero-form constraints on-shell. Two examples
thereof are a twisting of C[(Z_2)^3] that yields the original model, and the
Clifford algebra Cl_2n which provides an FCS formulation of the bosonic
Konstein--Vasiliev model with gauge algebra hu(4^{n-1},0).
