  Indefinite causal order (ICO) is playing a key role in recent quantum
technologies. Here, we experimentally study quantum thermodynamics driven by
ICO on nuclear spins using the nuclear magnetic resonance system. We realize
the ICO of two thermalizing channels to exhibit how the mechanism works, and
show that the working substance can be cooled or heated albeit it undergoes
thermal contacts with reservoirs of the same temperature. Moreover, we
construct a single cycle of the ICO refrigerator based on the Maxwell's demon
mechanism, and evaluate its performance by measuring the work consumption and
the heat energy extracted from the low-temperature reservoir. Unlike classical
refrigerators in which the coefficient of performance (COP) is perversely
higher the closer the temperature of the high-temperature and low-temperature
reservoirs are to each other, the ICO refrigerator's COP is always bounded to
small values due to the non-unit success probability in projecting the
ancillary qubit to the preferable subspace. To enhance the COP, we propose and
experimentally demonstrate a general framework based on the density matrix
exponentiation (DME) approach, as an extension to the ICO refrigeration. The
COP is observed to be enhanced by more than three times with the DME approach.
Our work demonstrates a new way for non-classical heat exchange, and paves the
way towards construction of quantum refrigerators on a quantum system.
