In this work, we study the dynamics of a single active Brownian particle, as well as the collective behavior of interacting active Brownian particles, in a fluctuating heterogeneous environment. We employ a variant of the diffusing diffusivity model where the equation of motion of the active particle involves a time-dependent motility and diffusivities. Within our model, those fluctuations are coupled to each other. Using analytical methods, we obtain the probability distribution function of particle displacement and its moments for a single particle. We then investigate the impact of the environmental fluctuations on the collective behavior of the active Brownian particles by means of extensive numerical simulations. Our results show that the fluctuations hinder the motility-induced phase separation, accompanied by a significant change of the density dependence of particle velocities. These effects are interpreted using our analytical results for the dynamics of a single particle.