We discuss cosmology based on a Cuscuta-Galileon gravity theory, which preserves just two degrees of freedom. Although there exist no additional degrees of freedom, introduction of a potential of a scalar field changes the dynamics. The scalar field is completely determined by matter fields. Giving an exponential potential as an example, we discuss the cosmological dynamics. The gravitational "constant"GF appeared in the effective Friedmann equation becomes time dependent. We also present how to construct a potential when we know the evolution of the Hubble parameter. When we assume the ΛCDM cosmology for the background evolution, we find the potential form. We then analyze the density perturbations, which equation is characterized only by a change of the gravitational "constant"Geff, which also becomes time dependent. From the observational constraints such as the constraint from the big-bang nucleosynthesis and the constraint on time-variation of gravitational constant, we restrict the parameters in our models. The time dependence of the gravitational constant in the effective Friedmann equation, we may have a chance to explain the Hubble tension problem.
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