Why can endothermic reactions occur spontaneously if the entropy change is negative for a cooling process?

An endothermic process is one which cools as the reaction proceeds meaning that the entropy change of the reaction will be negative. These reactions may still be feasible as the spontaneous nature of a reaction is based on a quantity measured in kJ/mole called the Gibbs free energy. It is defined by the change in enthalpy of the reaction minus the temperature in Kelvin multipled by the change in entropy. If the value is positive, then the reaction will not occur spontaneously without the input of energy. If the value of Gibbs free energy is negative or equal to zero then the reaction will proceed spontaneously. 

It is also important to note that the second law of thermodynamics (stating that everything tends towards disorder) is still valid for a process which decreases in entropy as long as the overall entropy change is positive. To apply this to the endothermic reaction, the heat given out by the reaction in order to decrease the temperature of the products is absorbed by the surrounding area whose entropy value will be positive by a greater value than the reaction is negative.