  We clarify the features of primordial non-Gaussianities of tensor
perturbations in Gao's unifying framework of scalar-tensor theories. The
general Lagrangian is given in terms of the ADM variables so that the framework
maintains spatial covariance and includes the Horndeski theory and
Gleyzes-Langlois-Piazza-Vernizzi (GLPV) generalization as specific cases. It is
shown that the GLPV generalization does not give rise to any new terms in the
cubic action compared to the case of the Horndeski theory, but four new terms
appear in more general theories beyond GLPV. We compute the tensor 3-point
correlation functions analytically by treating the modification to the
dispersion relation as a perturbation. The relative change in the 3-point
functions due to the modified dispersion relation is only mildly
configuration-dependent. When the effect of the modified dispersion relation is
small, there is only a single cubic term generating squeezed non-Gaussianity,
which is the only term present in general relativity. The corresponding
non-Gaussian amplitude has a fixed and universal feature, and hence offers a
"consistency relation" for primordial tensor modes in a quite wide class of
single-field inflation models. All the other cubic interactions are found to
give peaks at equilateral shapes.
