Efficiency of parallel and sequential heat pumps Consider two identical heat pumps, for example, split-system air conditioners. There're two ways to make them work - in parallel or sequential. Parallel means that "hot" radiators of the machines are put outside the room, while the "cold" radiators are put inside the room. Sequential means that the cold radiator of the first machine is inside the room, while the hot radiator is put inside, say, a large isolated box, where it is cooled by the cold radiator of the second machine, and the hot radiator of the second machine is outside the room.
In which case - parallel or sequential - the efficiency of the cooling would be better?
 A: Efficiency of one heat pump in cooling can be defined by expression
$$\eta = \frac{Q_C}{W},$$
that is heat that is taken from cooler reservoir divided by the work put into the pump.
If you have two pumps in parallel efficiency shall be the same, as you will have heat twice as large and work twice as large
$$\eta = \frac{2Q_C}{2W}.$$
If you however have two pumps in sequence equation reads as 
$$\eta = \frac{Q_C}{2W}.$$
So in order to obtain at least same efficiency you should extract heat twice as large by each pump.  So the efficiency of each pump should be twice as large for about half of the temperature change.  So far so easy.
In the next step we must take into consideration exact cyclical thermodynamical processes.  There are plenty of them that you can use and no can be exactly theoretically calculated.  In such cases it is useful to observe the most efficient thermodynamical process, that is Carnot cycle and extract conclusions from it.
Efficiency of the heat engine based on Carnot cycle can be shown to be
$$\eta = \frac{T_C}{T_H-T_C}.$$
In case of two pumps in the first iteration intermediate temperature $T' = \frac{T_C+T_H}{2}$ in exactly in the middle.  Efficiency of two pumps shall be
$$\eta_1 = \frac{T_C}{T'-T_C}, \eta_2 = \frac{T'}{T_H-T'}$$
Obviously $\eta_1 = 2 \eta$ and $\eta_2 > 2 \eta$, therefore two sequential pumps in cooling should be more efficient.
It is interesting that two sequential pumps in warming should be less efficient using the same arguments!  I cannot find the error in the reasoning, so before accepting the answer, please wait some time that others check and give their comments.
A: The COP of a heat pump is a function of temperature lift (change from source to destination).  The higher the lift the lower the COP.  30C lift at 300% COP a practical rule of thumb.  In paralel you can get twice the heat at a constant lift.  In series twice the lift for the same total heat generated at the same efficiency.  Higher (perhaps double) efficiency but the same total heat as a single heat pump can be achieved serially, by halving each's lift.
High lift is generally a limitation of heat pumps and so a serial arrangement is a method of overcomming those limitations.
