Why would it run hotter when the whole point of an egr is to recirculate hot air/smoke etc into the engine for better short journey warm up. Surely with ONLY cool air from the filter at the front , the cylinder head should get cooler air without crap from the egr opening ???
When warming up, the car is circulating warm air back into the engine, so a car with an operating EGR will warm up quicker.
In general running though, a car without an EGR will
run hotter (but not to a noticeable level!!). As you know, the EGR is designed to reduce NOx emissions, these are produced at higher temperatures, so the whole point of EGRs is to reduce combustion temperature (and efficiency!).
The EGR allows exhaust gas into the cylinder, which reduces the efficiency of the combustion (less O2 to burn), and so reduces the combustion temperature (for NOx) along with all the detrimental things we know about
: reduced fuel efficiency, less power, more soot production, more hydrocarbons down the exhaust, more swirl flap failures, more DPF failures, fouling of the manifold (and further reduction in efficiency) etc.
Basically, I wouldn't worry about the 'high temperature'.