Autosampler Temperature Won't Stay at 5C

We are running a method on an H-Class  that requires that autosampler temperature to be 2-8C. We have the method set at 5C +/- 3C. It has been a challenge to keep the temperature below 8C to prevent instrument faliures. Our lab temperature has been monitored between 21-23C at the time of failure, and I'm sure it's warmer by a few degrees by the LCs, but it doesn't seem like we should be over 28C (which would make ambient 20C above the target temperature). It has been suggested that we "thaw" everything out by increasing temperature to 35C and then decreasing to 5C, which does seem to help, but we are still running at about 6C without any vials in the autosampler. As soon as the door is opened to add samples the temperature increases to about 8-10C and then hovers there with it not always decreasing back down into the accepted range. Does anyone have any tricks on how we might improve the performance of our autosampler temperature? Would it be better if we keep the autosampler at ambient temperature until all samples are loaded and then decrease to 5C? Or are we better off loading previously refrigerated vials into the autosampler?