In this paper, we are interested in some questions of Greven and den Hollander about the rate function $I_{\eta}^q$ of quenched large deviations for random walk in random environment. By studying the hitting times of RWRE, we prove that in the recurrent case, $\lim_{\theta\to 0^+}(I_{\eta}^q)''(\theta )=+\infty$, which gives an affirmative answer to a conjecture of Greven and den Hollander. We also establish a comparison result between the rate function of quenched large deviations for a diffusion in a drifted Brownian potential, and the rate function for a drifted Brownian motion with the same speed.