In this paper, we study two different models of dark energy based on the Chaplygin gas equation of state. The first model is the variable modified Chaplygin gas, while the second one is the extended Chaplygin gas. Both models are considered in the framework of higher order \(f(R)\) modified gravity. We also consider the case of time-varying gravitational constant \(G\) and \(\Lambda \) for both models. We investigate some cosmological parameters such as the Hubble, the deceleration, and the equation of state parameters. Then we show that the model that we consider, the extended Chaplygin gas with time-dependent \(G\) and \(\Lambda \) , is consistent with the observational data. Finally we conclude with the discussion of cosmological perturbations of our model.

Motivated by the work of Yang et al. (Mod. Phys. Lett. A 26:191, 2011), we report on a study of the new holographic dark energy (NHDE) model with energy density given by \(\rho _D\!=\!\frac{3\phi ^2}{4\omega }(\mu H^2\!+\!\nu {\dot{H}})\) in the framework of chameleon Brans–Dicke cosmology. We have studied the correspondence between the quintessence, the DBI-essence, and the tachyon scalar-field models with the NHDE model in the framework of chameleon Brans–Dicke cosmology. Deriving an expression of the Hubble parameter \(H\) and, accordingly, \(\rho _D\) in the context of chameleon Brans–Dicke chameleon cosmology, we have reconstructed the potentials and dynamics for these scalar-field models. Furthermore, we have examined the stability for the obtained solutions of the crossing of the phantom divide under a quantum correction of massless conformally invariant fields, and we have seen that the quantum correction could be small when the phantom crossing occurs and the obtained solutions of the phantom crossing could be stable under the quantum correction. It has also been noted that the potential increases as the matter–chameleon coupling gets stronger with the evolution of the universe.