Studies document the fact that temperature changes strongly affect interactions in many consumer-resource systems through altered, or shifted, phenologies. The mistiming of events, such as migration or emergence times, or the contraction or expansion of development times can upset the normal synchronization and lead to increased or decreased predation events. In this paper, we formulate a continuous time, phenologically-structured model of predator-prey interactions that is driven by temperature variations. It is particularly applicable to arthropod interactions because their development rates are so strongly temperature related. The model takes the form of a system of partial differential-integral equations for the species' population densities in development-time variables. In special cases, the model is analytically tractable and we find a closed-form solution. By calculating density variations under different temperature regimes, the model gives a quantitative method for assessing the effects of global temperature change on consumer-resource interactions.