We study the phenomenology of a class of minimally modified gravity theories called $f(mathcal{H})$ theories, in which the usual general relativistic Hamiltonian constraint is replaced by a free function of it. After reviewing the construction of the theory and a consistent matter coupling, we analyze the dynamics of cosmology at the levels of both background and perturbations, and present a concrete example of the theory with a $3$-parameter family of the function $f$. Finally, we compare this example model to Planck data as well as some later-time probes, showing that such a realization of $f(mathcal{H})$ theories fits the data significantly better than the standard $Lambda$CDM model, in particular by modifying gravity at intermediate redshifts, $zsimeq743$.
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