Most contemporary mortality models rely on extrapolating trends or past events. However, population dynamics will be significantly impacted by climate change, notably the influence of temperatures on mortality. In this paper, we introduce a novel approach to incorporate temperature effects on projected mortality using a multi-population mortality model. This method combines a stochastic mortality model with a climate epidemiology model, predicting mortality variations due to daily temperature fluctuations, be it excesses or insufficiencies. The significance of this approach lies in its ability to disrupt mortality projections by utilizing temperature forecasts from climate models and to assess the impact of this unaccounted risk factor in conventional mortality models. We illustrate this proposed mortality model using French data stratified by gender, focusing on past temperatures and mortality. Utilizing climate model predictions across various IPCC scenarios, we investigate gains and loss in life expectancy linked to temperatures and the additional mortality induced by extreme heatwaves, and quantify them by assessing this new risk factor in prediction intervals. Furthermore, we analyze the geographical differences across the Metropolitan France.