Quantum two-state systems, known as quantum bits (qubits), are unavoidably in contact with their uncontrolled thermal environment, also known as a macroscopic 'bath'. The higher the temperature of the qubits, the more impure their quantum state and the less useful they are for coherent control or quantum logic operations, hence the desirability of cooling down the qubits as much and as fast as possible, so as to purify their state prior to the desired operation. Yet, the limit on the speed of existing cooling schemes, which are all based on Markovian principles, is either the duration of the qubit equilibration with its bath or the decay time of an auxiliary state to one of the qubit states. Here we pose the conceptual question: can one bypass this existing Markovian limit? We show that highly frequent phase shifts or measurements of the state of thermalized qubits can lead to their ultrafast cooling, within the non-Markov time domain, well before they re-equilibrate with the bath and without resorting to auxiliary states. Alternatively, such operations may lead to the cooling down of the qubit to arbitrarily low temperatures at longer times. These anomalous nonMarkov cooling processes stem from the hitherto unfamiliar coherent quantum dynamics of the qubit-bath interaction well within the bath memory time.