  Quantum speed limit (QSL) for open quantum systems in the non-Markovian
regime is analyzed. We provide a the lower bound for the time required to
transform an initial state to a final state in terms of thermodynamic
quantities such as the energy fluctuation, entropy production rate and
dynamical activity. Such bound was already analyzed for Markovian evolution
satisfying detailed balance condition. Here we generalize this approach to deal
with arbitrary evolution governed by time-local generator. Our analysis is
illustrated by three paradigmatic examples of qubit evolution: amplitude
damping, pure dephasing, and the eternally non-Markovian evolution.
