Acute hospital care as performed in the intensive care unit (ICU) is characterized by its frequent, but short-term interventions for patients who are severely ill. Because clinicians have to attend to more than one patient at a time and make decisions in a limited time in acute hospital care environments, the accurate prediction of the in-hospital mortality risk could assist them to pay more attention to patients with a higher in-hospital mortality risk, thereby improving the quality and efficiency of the care. One of the salient features of ICU is the diversity of patients: clinicians are faced by patients with a wide variety of diseases. However, mortality prediction for ICU patients has typically been conducted by building one common predictive model for all the diseases. In this paper, we incorporate disease-specific contexts into mortality modeling by formulating the mortality prediction problem as a multi-task learning problem in which a task corresponds to a disease. Our method effectively integrates medical domain knowledge relating to the similarity among diseases and the similarity among Electronic Health Records (EHRs) into a data-driven approach by incorporating graph Laplacians into the regularization term to encode these similarities. The experimental results on a real dataset from a hospital corroborate the effectiveness of the proposed method. The AUCs of several baselines were improved, including logistic regression without multi-task learning and several multi-task learning methods that do not incorporate the domain knowledge. In addition, we illustrate some interesting results pertaining to disease-specific predictive features, some of which are not only consistent with existing medical domain knowledge, but also contain suggestive hypotheses that could be validated by further investigations in the medical domain.