Rumours on social media exhibit complex temporal patterns. This paper develops a model of rumour prevalence using a point process, namely a log-Gaussian Cox process, to infer an underlying continuous temporal probabilistic model of post frequencies. To generalize over different rumours, we present a multi-task learning method parametrized by the text in posts which allows data statistics to be shared between groups of similar rumours. Our experiments demonstrate that our model outperforms several strong baseline methods for rumour frequency prediction evaluated on tweets from the 2014 Ferguson riots.