The behaviour of information cascades (such as retweets) has been modelled extensively. While point process-based generative models have long been in use for estimating cascade growths, deep learning has greatly enhanced diverse feature integration. We observe two significant temporal signals in cascade data that have not been emphasized or reported to our knowledge. First, the popularity of the cascade root is known to influence cascade size strongly; but the effect falls off rapidly with time. Second, there is a measurable positive correlation between the novelty of the root content (with respect to a streaming external corpus) and the relative size of the resulting cascade. Responding to these observations, we propose GammaCas, a new cascade growth model as a parametric function of time, which combines deep influence signals from content (e.g., tweet text), network features (e.g., followers of the root user), and exogenous event sources (e.g., online news). Specifically, our model processes these signals through a customized recurrent network, whose states then provide the parameters of the cascade rate function, which is integrated over time to predict the cascade size. The network parameters are trained end-to-end using observed cascades. GammaCas outperforms seven recent and diverse baselines significantly on a large-scale dataset of retweet cascades coupled with time-aligned online news -- it beats the best baseline with an 18.98% increase in terms of Kendall's $τ$ correlation and $35.63$ reduction in Mean Absolute Percentage Error. Extensive ablation and case studies unearth interesting insights regarding retweet cascade dynamics.