Changes in temperature and rainfall will lead to frequent occurrence of floods, droughts as well as heat and cold waves. In this paper we introduce a new generalized counting process with Gumbel inter-arrival time distribution. This count model can be applied to analyze the climate variability due to the changes in temperature and rainfall. The decreasing hazard function of the new model leads to over dispersion, whereas increasing hazard function leads to underdispersion. Thus this new Gumbel count model can model both over and under dispersed count data. The new model has many nice features such as its closed form nature, computational simplicity, ability to nest Poisson, existence of moments etc. The use of the model is illustrated in two applications with respect to a data on the monthly rainfall of Kerala, the southern state of India which crossed the extreme level during 2005-2008 (under dispersed) and the data on the daily temperature in Kerala which attained the maximum level in Kerala during 2005-2008 (over dispersed).