Does a warmer body and warmer surroundings cool faster than a cooler body and surrounding with equal difference in temperature? Consider this.
Sphere 1 = 20 degree celcius
Surrounding 1 = 5 degree celcius
Sphere 2 = 90 degree celcius
Surrounding 2 = 75 degree celcius
The two above is in two whole different systems, does not effect each other.
Heat flow goes from warm to cold, and more flow if a object is for example much warmer than its surroundings. But if the temperature difference are the same but one system is warmer than the other, does this create a difference in heat transfer between them?  
 A: All other things equal, in the cases of conduction and convection, the rate of heat transfer between two objects is proportional to the temperature difference, $T_{2}-T_{1}$. So for these cases the rates should be the same.,
For radiation the rate is proportional to the difference in the temperatures to the fourth power or $T_{2}^{4}-T_{1}^{4}$. So for radiation the rate of heat transfer will be greater for the warmer bodies.
Just to be clear, the above is for comparing the difference in heat transfer rates for a each given transfer mechanism. The overall heat transfer rate will be determined by the actual combination of mechanisms involved, and which mechanism(s) dominates, for a particular scenario. 
Hope this helps.
A: Newton's law of cooling states that the rate of heat loss from a body is proportional to the difference in temperature between the body and its surroundings.  
This is not an exact formulation but produces a reasonable approximation if the temperature differences are small and if the mechanisms of heat transfer do not change.  
If the transfer is due to thermal conduction then the relationship works reasonably well as long as the coefficient of thermal conductivity is not a rapidly varying function of temperature.  
With thermal convection being the transfer mechanism the law is only approximately true but is better if there is forced convection.  
Loss by thermal radiation which depends on the fourth power of the temperature in kelvin will follow the law only very approximately and then only if the differences in temperature are very small.
