We study the multi-fractional theory with $q$ -derivatives, where the
multi-fractional measure is considered to be in the time direction. The
evolution of power spectra and also the expansion history of the universe are
investigated in the $q$ -derivatives theory. According to the matter power
spectra diagrams, the structure growth would increase in the multi-fractional
model, expressing incompatibility with low redshift measurements of large scale
structures. Furthermore, concerning the diagrams of Hubble parameter evolution,
there is a reduction in the value of Hubble constant which conflicts with local
cosmological constraints. We also explore the multi-fractional model with
current observational data, principally Planck 2018, weak lensing, supernovae,
baryon acoustic oscillations (BAO), and redshift-space distortions (RSD)
measurements. Numerical analysis reveals that the degeneracy between
multi-fractional parameters makes them remain unconstrained under observations.