Abstract: A resilient lid insert or membrane construction is outfitted upon a lidded beverage container for enabling the user to transfer heat from a relatively hot assembly-contained beverage prior to consumption. The insert or membrane construction may include a resilient material and may optionally provide a beverage-damming structure. The membrane construction defines beverage-containing and beverage-cooling compartments. The membrane construction may include at least one aperture for letting matter intermediate the beverage-containing and beverage-cooling compartments. The beverage-cooling compartment receives heat from the beverage received therein and thereby enables the beverage to cool before being outlet via a primary outlet. The resilient membrane construction may be biasable intermediate a relaxed and an actuated configuration.