Abstract In this paper, we consider fractional ordinary differential equations with not instantaneous impulses. Firstly, we construct a uniform framework to derive a formula of solutions for impulsive fractional Cauchy problem involving generalization of classical Caputo derivative with the lower bound at zero. In other words, we mean a different solution keeping in each impulses the lower bound at zero, which can better characterize the memory property of fractional derivative. Secondly, we introduce a new concept of generalized Ulam–Hyers–Rassias stability. Then, we choose a fixed point theorem to derive a generalized Ulam–Hyers–Rassias stability result for such new class of impulsive fractional differential equations. Finally, an example is given to illustrate our main results.