Pemetrexed Disodium and Cisplatin With or Without Cediranib Maleate in Treating Patients With Malignant Pleural Mesothelioma

Inclusion Criteria: * Patient must have histologically or cytologically confirmed diagnosis of malignant pleural mesothelioma; surgical resection must not be planned * Patients must have measurable or non-measurable disease by both RECIST and modified RECIST criteria for pleural tumors as documented by computed tomography (CT) scan; examinations for assessment of measurable disease must have been completed within 28 days prior to registration; examinations for assessment of non-measurable disease must have been performed within 42 days prior to registration; all disease must be assessed and documented on the RECIST 1.1 and Modified RECIST Baseline Tumor Assessment Form * Patients must not have received any prior systemic therapy (chemotherapy or other biologic therapy) for their unresectable malignant pleural mesothelioma; prior systemic chemotherapy or biologic therapy is allowed as neoadjuvant or adjuvant treatment, disease has now recurred, and all systemic treatment was completed \> 6 months prior registration; prior therapy must not have included cediranib * Patients may have received prior surgery (e.g., pleurectomy, pleurodeisis) provided at least 28 days have elapsed since surgery (thoracic or other major surgeries) and patients have recovered from all associated toxicities at the time of registration; there must be no anticipated need for major surgical procedures during protocol treatment * Patients may have received prior radiation therapy provided at least 28 days have elapsed since the last treatment and patients have recovered from all associated toxicities at the time of registration * Institutions must seek additional patient consent for the banking and future use of specimens * Patient must have Zubrod performance status 0-2 * Absolute neutrophil count \>= 1,500 mcl * Platelets \>= 100,000/ml * Hemoglobin \>= 9.0 g/dl (may be reached by transfusion) * Total bilirubin =\< 1.5 x institutional upper limit of normal (IULN) * Serum glutamic oxaloacetic transaminase (SGOT) or serum glutamate pyruvate transaminase (SGPT) =\< 2.5 x ULN (if liver metastases are present, SGOT or SGPT must be =\< 5.0 x IULN) * Serum creatinine =\< 1.5 x IULN * Calculated creatinine clearance \>= 60 mL/min * Urine protein must be screened by urine analysis within 28 days prior to registration; patient must not have greater than +1 proteinuria on two consecutive dipsticks taken no less than 7 days apart; however, if the first urinalysis shows no protein, then a repeat urinalysis is not required * Patient must have an electrocardiogram (ECG) performed within 42 days prior to registration; patient must not have mean corrected QT (QTc) \> 500 msec (with Bazett's correction) in screening electrocardiogram, or other significant ECG abnormality, New York Heart Association (NYHA) classification III or IV; patient must not require concurrent use of drugs or biologics with proarrhythmic potential * Patient must not be receiving any medication that may markedly affect renal function (e.g., vancomycin, amphotericin, pentamidine) * Patient must not have had clinically significant hemoptysis, defined as greater than 1 tablespoon of bright red blood, within one year prior to registration; although hemoptysis has not been associated with cediranib, it may be a class effect for anti-angiogenic agents and therefore a risk factor for this experimental agent * Patient must be able to swallow oral medications * Patients must not have known human immunodeficiency virus (HIV) infection * Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures * Patient must have no plans to receive concurrent chemotherapy, hormonal therapy, radiotherapy, immunotherapy or any other type of therapy for treatment of cancer while on this protocol treatment * No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for five years