As climate change intensifies, disparities in people's heat exposure are emerging as a critical public health concern, including in wealthy countries like Switzerland. This study investigates spatial and socio-demographic differences in outdoor heat exposure across 1625 Swiss municipalities, using satellite data and predicted air temperature data within a multi-dimensional heat exposure framework encompassing a composite heat exposure index (CHEI) combining heat intensity, heatwave duration, and historical heatwave probability. Using stepwise weighted least squares (WLS) regression models, we first model socio-demographic predictors, then add topography, and finally incorporate urban-form variables to assess heat exposure disparities associated with economic status, age structure, immigration background, social assistance, and living conditions. We further use geographically weighted regression (GWR) to capture spatial heterogeneity and classify municipalities as overexposed, underexposed, or showing no significant disparity. Our findings reveal that high-income municipalities tend to experience higher heat exposure. At the same time, municipalities with larger shares of non-EU foreigners and residents receiving social assistance are also more exposed than others, underscoring the intersection of heat risk with socially marginalized and affluent communities in larger cities. Yet many of these associations weaken after controlling for elevation and urbanization, highlighting the critical role of physical geography in the Swiss context. For age structure, regression models suggest weak or negative associations between elderly concentration and heat exposure after accounting for physical factors; however, quartile analyses reveal that municipalities with higher shares of residents aged over 80 still face higher exposure in certain regions. Our findings emphasize the need to address socio-demographic heat disparities in affluent societies with diverse population structures, large aging population, where topography and degree of urbanisation can amplify local heat burdens. Integrating social vulnerability with geographic and morphological drivers is therefore essential for designing targeted adaptation measures and reducing unequal heat risks.