I used to live in a cool Europan type climate with snow in winter and very temperate summers. I was full of energy and zip and go. I then moved into the tropics. In this hot and humid climate I feel like I have aged by at least 20 years and can barely motivate myself to do anything, neither physically nor mentally. I had to install air con throughout my house to be at least 50% productive during the hot summer months.

So, yes. Climate does influence productivity, both physical and mental.