The temperature dependence of domain-wall depinning in permalloy nanowires is investigated by measuring depinning fields and corresponding depinning times as a function of the external magnetic bias field. Domain walls are pinned at triangular notches in the nanowires and detected noninvasively by Hall micromagnetometry. This technique allows one to acquire depinning-field and depinning-time distributions in the temperature range between 5 and 50K and thus to determine the stochastics of the depinning process. The results are discussed in terms of the NéelBrown model for thermally activated magnetization reversal, assuming a single energy barrier to overcome. In general, the cases presented deviate from this description and give a clear indication that a more complex term for the energy landscape of domain-wall depinning at constrictions in nanowires is obligatory.