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In this work, we introduce two models of the hybrid metric-Palatini theory ofgravitation. We explore their background evolution, showing explicitly that onerecovers standard General Relativity with an effective Cosmological Constant atlate times. This happens because the Palatini Ricci scalar evolves towards andasymptotically settles at the minimum of its effective potential duringcosmological evolution. We then use a combination of cosmic microwavebackground, supernovae and baryonic accoustic oscillations background data toconstrain the models' free parameters. For both models, we are able toconstrain the maximum deviation from the gravitational constant $G$ one canhave at early times to be around $1\%$.

Constraints on hybrid metric-Palatini models from background evolution