A reliable online speaking rate estimation tool is useful in many domains, including speech recognition, speech therapy intervention, speaker identification, etc. This paper proposes an online speaking rate estimation model based on recurrent neural networks (RNNs). Speaking rate is a long-term feature of speech, which depends on how many syllables were spoken over an extended time window (seconds). We posit that since RNNs can capture long-term dependencies through the memory of previous hidden states, they are a good match for the speaking rate estimation task. Here we train a long short-term memory (LSTM) RNN on a set of speech features that are known to correlate with speech rhythm. An evaluation on spontaneous speech shows that the method yields a higher correlation between the estimated rate and the ground-truth rate when compared to the state-of-the-art alternatives. The evaluation on longitudinal pathological speech shows that the proposed method can capture long-term and short-term changes in speaking rate.