Pembrolizumab With or Without Anetumab Ravtansine in Treating Patients With Mesothelin-Positive Pleural Mesothelioma

Inclusion Criteria: * PRE-REGISTRATION * Patients must have histologically or cytologically confirmed malignant pleural mesothelioma * Patient is willing to submit a tissue sample to test for expression of mesothelin * Note: Tissue sample for mesothelin assay may have been collected prior to, during or after receipt of the frontline chemotherapy; patients will not be required to submit another tissue sample after receipt of the chemotherapy * Patients must have received platinum based chemotherapy * REGISTRATION * For phase 2 only: * Patient has measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 for non-pleural disease or modified RECIST 1.1 (mRECIST) for pleural disease * Note: For pleural disease, this is defined as at least one lesion that can be accurately measured perpendicular to the chest wall or mediastinum that is \>= 10 mm (\>= 1 cm); for extra pleural disease, measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 10 mm (\>= 1 cm) for non-nodal lesions and \>= 15 mm (\>= 1.5 cm) for nodal lesions with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam as per RECIST 1.1 * Tissue submitted for testing at pre-registration shows moderate or stronger mesothelin expression in \>= 30% of the tumor cells * For phase 1 and 2: * Patients mu…