Yes, the EJ253 has a built-in oil cooler that mounts between the block and the oil filter. The problem is that oil temp is changed by the engine coolant only, not moving air. That means the cooler actually functions better as an oil heater when the engine is warming up, than a cooler when the t-stat is open and coolant is flowing through the whole system.
An external filter with electric fan and oil t-stat has proven much more effective in keeping oil, and thus coolant, temps down. Proof is that many of the EJ253 engine owners have removed the stock oil cooler and replaced them with external coolers with fans.
In addition, a car with marginal overheating will operate at higher cooling temps with the internal cooler, since oil temps are normally 20-30 degrees F more than the coolant temp. Result: instead of cooling oil, you are only raising coolant temps. Problem eliminated with quality, external oil cooler. I installed a Setrab 1-series, 19 row with fan and shroud, the largest I had room for.