  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,
$z\simeq743$.
