Hate speech has become a pressing issue on social media platforms, inflicting significant harm on users. It encompasses a wide range of targets, such as misogyny, sexism, racism, xenophobia, homophobia, and more. Each form of hate speech specifically targets vulnerable groups based on their unique characteristics. To timely prevent the spread of hate speech, existing research employs machine learning and deep learning methods to predict whether the level of hatred in ongoing conversations will escalate. These methods typically involve feeding an entire conversation into a hate trend prediction model, which then assesses whether the overall level of hostility within the conversation is likely to escalate However, previous studies cast the problem into a binary classification task without addressing either the target-oriented nature of hate speech. In this paper, we propose Target-specific hAte tRend PredictiOn Network(TARPON), which is an end-to-end model. TARPON filters content unrelated to the target we are interested in through Multi-Granularity Target Embeddings Extraction Module, focusing only on predicting specific target trend. Experiments on three real-world conversation datasets show TARPON is effective and outperforms state-of-the-art baselines.