Is the phrase "on that day" correct like that, if not how do I put it in the sentence ??

Abraham Lincoln once said, "Most folks are about as happy as they make up their minds to be." 

Your sentence reminded me of that quote, because I think you are trying to say that they did just that. Technically, I don't think your sentence is wrong, just awkward.  Hot weather implies that it was probably sunny, as rain would cool things off.  The way you have this worded seems to be saying that dealing with the hot weather is what might lead them to be happy.  I suspect that your meaning is a bit different.  Does the following work?

They chose to embrace the happiness of the day rather than get side-tracked by the terribly hot weather.