Disformal transformations in a Palatini extension of Horndeski's gravity

In this paper, we extend Horndeski's theory into the Palatini approach, assuming that the metric tensor and the (symmetric) connection are a priori independent objects. We introduce an additional transformation of the connection and write down the action functional being form-invariant under both the disformal transformation of the metric and the new transformation of the connection. We show that such a theory reduces on-shell to a metric subclass of Horndeski's gravity called kinetic gravity braiding. We also introduce an invariant metric and connection, and demonstrate that quantities defined in such a way lead to a metric theory. In the second part of the paper, we consider a simple cosmological model within the theory and explore its potential links with $k$-essence-type theories, with a non-trivial coupling between the scalar field and the matter part of the action in the Einstein frame. We show that there exists a model that reproduces late-time cosmic acceleration, approaching asymptotically the de Sitter phase, motivating further study of the theories.