Godbillon-Vey type functional for almost contact manifolds

Many contact metric manifolds are critical points of curvature functionals restricted to spaces of associated metrics. The Godbillon-Vey functional has never been considered in a variational context in contact geometry. Recently we extended this functional from foliations to arbitrary plane fields on a 3-dimensional manifold, so, the following question arises: can one use the Godbillon-Vey functional to find optimal almost contact manifolds? In the paper, we introduce a Godbillon-Vey type functional for a 3-dimensional almost contact manifold and find its Euler-Lagrange equations for all variations preserving the Reeb vector field. We construct critical (for our functional) 3-dimensional almost contact manifolds having a double-twisted product structure, these solutions belong to the class $C_{5}\oplus C_{12}$ according to Chinea-Gonzalez classification.