Not sure why that would happen. There is nothing in the theme itself which should cause this. We ran all of our demo sites on shared hosting like hostgator for a long time without issues. We did change to vps since resource limit reached its peak sometimes when we release a new theme due to extraordinary traffic spike seen when a new theme gets released on ThemeForest WordPress themes page. And of course our needs got bigger due to increase in number of themes and some of these were multi-purpose which lots of data and Appdev is not one of those. We have not had anyone report this to us for any of our themes to be honest.
How much traffic do you have on this site? Even if the theme code is really the issue, you can enable caching (WP Super Cache recommended) and the problem should go away.