Thanks for your comment. That is exactly the mechanism that is believed to be driving the relation between high temperatures and violence: High temperatures reduce a country's growth rate, which according to the empirical evidence increases the risk of violence. However, the link that is currently studied is the one between high temperatures and agricultural growth (harvests that falter because of droughts or excessive rains). But you are correct, there is likely also a link between high temperatures and productivity. Having lived in some very hot countries, I can only support your claim.