Average is one metric for assessing how good your dice were, hotness is just another. For average, the difference between rolling a 6 instead of a 7 is the same as that of a 2 instead of a 3, but for hotness, the latter is considered much better. If you roll exactly the expected number of each DR value (2 thru 12), the hotness score will be 0. Positive scores mean your dice were hot, negative means not so hot. The purpose of this feature is to give me some sort of pseudo-scientific basis for blaming the dice when I lose
A few more examples, from the log files you sent me.
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In this one, Alice rolled more 2's than expected (7, expected 3.9), which bumps the hotness score up. 3's and 4's and 5's were almost exactly as expected, and so don't affect the score, but there were more 6's than expected (25, expected 19.5), which will increase the score (but not by as much as the 2's, since 6's are considered less important than 2's). There were also less 8's, 10's and 12's than expected, which also increases the score. More 9's were rolled than expected, so this reduces the score by a bit. Final score +0.7, so somewhat hot.
Likewise, Bob rolled fewer 2's, 4's and 5's than expected, so his score goes down by quite a bit. He rolled fewer 8's, but more 9's, so they just about cancel each other out. His 10 thru 12's were almost exactly as expected, so have no effect on the score. Final score -0.3, so his dice weren't particularly good, but not tragically so.
This one is even more dramatic:
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Bob rolled a crazy amount of 3's (14, expected 4.5) and his 2's were pretty good as well (5, expected 2.2). These increased his score by a lot, and while his 4's and 5's were low, they don't reduce his score by as much because 2's and 3's are considered to be more important than 4's and 5's. On the other side, he rolled fewer 8's, 11's and 12's than expected, which increased his score by even more. He rolled more 10's than expected, which will decrease his score, but overall, he scored +6.2, which is pretty damn hot.