  The extension of quantum thermodynamics to situations that go beyond standard
thermodynamic settings comprises an important and interesting aspect of its
development. One such situation is the analysis of the thermodynamic
consequences of structured environments that induce a non-Markovian dynamics.
We study a quantum Otto refrigerator where the standard Markovian cold
reservoir is replaced by a specific engineered cold reservoir which may induce
a Markovian or non-Markovian dynamics on the quantum refrigerant system. The
two dynamical regimes can be interchanged by varying the coupling between the
refrigerant and the reservoir. An increase of non-Markovian effects will be
related to an increase of the coupling strength, which in turn will make the
energy stored in the interaction Hamiltonian, the interaction energy,
increasingly relevant. We show how the figures of merit, the coefficient of
performance and the cooling power, change for non-negligible interaction
energies, discussing how neglecting this effect would lead to an overestimation
of the refrigerator performance. Finally, we also consider a numerical
simulation of a spin quantum refrigerator with experimentally feasible
parameters to better illustrate the non-Markovian effects induced by the
engineered cold reservoir. We argue that a moderate non-Markovian dynamics
performs better than either a Markovian or a strong non-Markovian regime of
operation.
