Adding more oil makes the air/fuel ratio leaner. The terminology can be a bit confusing; a richer gas/oil ratio (more oil, lower ratio) makes the air/fuel ratio leaner (less fuel for the same airflow, higher ratio) with all else the same.
Since adding more oil leans the air/fuel ratio, it will generally make the engine run hotter (assuming that there was sufficient lubrication at the previous mixture).
I say generally because the highest combustion temperature in an engine occurs a bit richer than the stoichiometric air/fuel ratio. Since small engines usually run at a richer ratio than stoichiometric, the combustion temperature will increase as the ratio is leaned approaching the ratio yielding maximum temperature, then decrease as the ratio is further leaned.
TLDR: adding more oil to the mix will generally make a small two stroke engine run hotter not cooler with all else the same.