This study presents a comprehensive analysis of the characteristics of precipitation, temperature, and groundwater recharge in the recharge zone of the nine basins of the San Antonio segment of the Edwards Balcones Fault Zone Aquifer, which is one of the major groundwater systems in the United States and serves as primary water sources for approximately 1.7 million people in south‐central Texas. Datasets of monthly precipitation and average temperature (1895–2019) and groundwater recharge (1934–2019) are used to examine the decadal variability in precipitation, temperature, and groundwater recharge on the annual scale with a normalized 20‐year moving average of variance. Climate elasticity (precipitation and potential evapotranspiration) of groundwater recharge is estimated to evaluate impacts of climate change on groundwater recharge. The results of this study show that precipitation and temperature variability exhibit decadal cyclic patterns. Elasticity analysis of groundwater recharge indicates that a 1% change in annual precipitation may result in 2%, with a likely range of 0.15%–2.8%, change in groundwater recharge, and a 1% change in annual potential evapotranspiration may lead to −3.3% change in groundwater recharge with a likely range of −8.9% to 4% in the study area. This study suggests that climate elasticity of groundwater recharge may provide an alternative means for evaluating climate impacts on groundwater recharge to an aquifer.