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Observations and GCMs exhibit approximate proportionality between cumulative carbon dioxide (CO) emissions and global warming. Here we identify sufficient conditions for the relationship between cumulative CO emissions and global warming to be independent of the path of CO emissions; referred to as "path independence". Our starting point is a closed form expression for global warming in a two-box energy balance model (EBM), which depends explicitly on cumulative emissions, airborne fraction and time. Path independence requires that this function can be approximated as depending on cumulative emissions alone. We show that path independence arises from weak constraints, occurring if the timescale for changes in cumulative emissions (equal to ratio between cumulative emissions and emissions rate) is small compared to the timescale for changes in airborne fraction (which depends on CO uptake), and also small relative to a derived climate model parameter called the damping-timescale, which is related to the rate at which deep-ocean warming affects global warming. Effects of uncertainties in the climate model and carbon cycle are examined. Large deep-ocean heat capacity in the Earth system is not necessary for path independence, which appears resilient to climate modeling uncertainties. However long time-constants in the Earth system carbon cycle are essential, ensuring that airborne fraction changes slowly with timescale much longer than the timescale for changes in cumulative emissions. Therefore path independence between cumulative emissions and warming cannot arise for short-lived greenhouse gases.