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, $z\simeq 743$.