Markovian behavior involves a rather complicated cooperation between system and bath dynamics. This means that in phenomenological treatments, one cannot combine arbitrary system Hamiltonians, H S , with a given L-GKS generator. This observation is particularly important in the context of quantum thermodynamics, where it is tempting to study Markovian dynamics with an arbitrary control Hamiltonian. Erroneous derivations of the quantum master equation can easily lead to a violation of the laws of thermodynamics.