Cediranib Maleate and Olaparib in Treating Patients With Recurrent Ovarian, Fallopian Tube, or Peritoneal Cancer or Recurrent Triple-Negative Breast Cancer

Inclusion Criteria:

* PHASE I: Participants must have histologically or cytologically confirmed epithelial ovarian cancer, primary peritoneal serous cancer, fallopian tube cancer, or triple-negative breast cancer
* PHASE II: Participants must have histologically or cytologically grade 2 or 3 (high-grade) papillary-serous or endometrioid epithelial ovarian cancer, primary peritoneal serous cancer, or fallopian tube cancer; participants with epithelial ovarian, primary peritoneal, or fallopian tube cancers of other high-grade histologies who carry a known deleterious breast cancer gene (BRCA) germline mutation by standard clinical testing (Myriad BRAC Analysis) will also be considered eligible
* PHASE I-T: Participants must have histologically or cytologically confirmed epithelial ovarian cancer, primary peritoneal serous cancer, or fallopian tube cancer
* Ovarian cancer, primary peritoneal, and fallopian tube participants in the Phase 1 and Phase 1-T portions of this trial must have either measurable cancer by RECIST 1.1 criteria or an elevated cancer antigen (CA)125 level at least twice the upper limit of normal on two separate occasions at least 1 day but not more than 3 months apart; at least one of the samples should be within 1 week of starting treatment; patients with both an elevated CA125 and measurable cancer will be followed by RECIST 1.1 criteria; patients with only an elevated CA125 level will be followed by modified Gynecologic Cancer Intergroup (GCIG) criteria
* Participants in the Phase II portion of the trial must have measurable disease by RECIST 1.1 criteria
* Breast cancer participants must have measurable disease by RECIST criteria
* PRIOR THERAPY PHASE I and PHASE I-T:

  * Prior chemotherapy for ovarian cancer patients must have included a first-line platinum-based regimen with or without intravenous consolidation chemotherapy
  * Breast cancer patients must have recurred post both an Adriamycin- and taxane-containing regimen
  * Prior hormonal-based therapy for ovarian, primary peritoneal serous, fallopian tube cancer, or breast cancer is acceptable
  * Patients may not have had a prior PAR polymerase (PARP)-inhibitor in the recurrent or metastatic setting; prior treatment with BSI-201 (iniparib) is allowed
  * Patients may not have had a prior anti-angiogenic agent in the recurrent or metastatic setting
* PRIOR THERAPY PHASE II:

  * Prior chemotherapy must have included a first-line platinum-based regimen with or without intravenous consolidation chemotherapy
  * Prior hormonal-based therapy for ovarian, primary peritoneal serous, or fallopian tube cancer is acceptable
  * Patients may not have previously received a PARP-inhibitor; prior treatment with BSI-201 is allowed
  * Patients may not have had a prior anti-angiogenic agent in the recurrent setting
  * Patients may have received up to 1 non-platinum-based line of therapy in the recurrent setting
  * Patients may have received an unlimited number of platinum-based therapies in the recurrent setting
  * Patients should have platinum-sensitive disease, where platinum-sensitive disease is defined as having had a \> 6 month interval since last receiving platinum therapy prior to disease recurrence; patients must have had a prior response while on the platinum-containing regimen and cannot have experienced disease progression while receiving platinum
* Subjects may begin cediranib and olaparib at least 3 weeks after their last dose of chemotherapy or hormonal therapy, assuming they are otherwise eligible
* Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of cediranib or olaparib in participants \< 18 years of age, children are excluded from this study but will be eligible for future pediatric trials
* Estimated life expectancy of greater than 6 months
* Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 (Karnofsky \> 60%)
* Absolute neutrophil count \>= 1,500/mcL
* Platelets \>= 100,000/mcL
* Hemoglobin \> 9 g/dL

  * For patients enrolled to the Phase 1-T portion of the protocol, the hemoglobin should be \>= 10 g/dL
* Total bilirubin within 1.5 times the upper limit of normal institutional limits
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal
* Creatinine =\< the institutional upper limit of normal or creatinine clearance \>= 60 mL/min/1.73 m\^2 for subjects with creatinine levels above institutional normal
* Less than or equal to 1+ proteinuria on two consecutive dipsticks taken no less than 1 week apart, or \< 1 gm protein on 24-hour urine collection or a urine protein: creatinine ratio of \< 1
* Troponin T or I within normal institutional limits
* Coagulation parameters (international normalized ratio \[INR\], activated partial thromboplastin time \[aPTT\]) within 1.25 x upper limit of normal institutional limits, except where a Lupus anti-coagulant has been confirmed
* Toxicities of prior therapy (except alopecia) should be resolved to less than or equal to grade 1 as per National Cancer Institute Common Terminology Criteria for Adverse Events version 4.0 (NCI-CTCAE v4.0); patients with long-standing stable grade 2 neuropathy may be considered after discussion with the overall principal investigator (PI)
* Subjects with treated limited stage basal cell or squamous cell carcinoma of the skin or carcinoma in situ of the breast or cervix are eligible; subjects with prior cancer treated with a curative intent with no evidence of recurrent disease 5 years following diagnosis and judged by the investigator to be at low risk of recurrence are eligible; subjects with any other concomitant or prior invasive malignancies are ineligible
* Patients who have the following risk factors are considered to be at increased risk for cardiac toxicities; these patients should have increased monitoring:

  * Prior treatment with anthracyclines
  * Prior treatment with trastuzumab
  * A New York Heart Association classification of II controlled with treatment
  * Prior central thoracic radiation therapy (RT), including RT to the heart
  * History of myocardial infarction within 12 months (patients with history of myocardial infarction within 6 months are excluded from the study)
* The effects of cediranib and olaparib on the developing human fetus are unknown; for this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 3 months following treatment discontinuation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately
* Ability to understand and the willingness to sign a written informed consent
* Patients must be able to tolerate oral medications and not have gastrointestinal illnesses that would preclude absorption of cediranib or olaparib
* Patients must be willing and able to check and record daily blood pressure readings

Exclusion Criteria:

* Participants who have had chemotherapy or radiotherapy within 3 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 3 weeks earlier
* Participants may not be receiving any other investigational agents nor have participated in an investigational trial within the past 4 weeks; subjects may not have received prior treatment affecting the vascular endothelial growth factor (VEGF) pathway in the recurrent setting, including thalidomide, bevacizumab, sunitinib, or sorafenib; in the Phase I portion of the trial, subjects may not have received prior treatment with oregovomab (OvaRex) or any other antibodies that may interfere with CA-125 measurements
* Patients with untreated brain metastases, spinal cord compression, or evidence of symptomatic brain metastases or leptomeningeal disease as noted on computed tomography (CT) or MRI scans should not be included on this study, since neurologic dysfunction may confound the evaluation of neurologic and other adverse events; screening imaging to rule out brain metastases is not required for screening, but should be performed prior to study enrollment if clinically indicated; patients with treated brain metastases and resolution of any associated symptoms must demonstrate stable post-therapeutic imaging for at least 6 months following therapy prior to starting study drug
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to cediranib maleate or olaparib
* Participants receiving any medications or substances that are strong inhibitors or inducers of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) are ineligible; dihydropyridine calcium-channel blockers are permitted for management of hypertension
* Patients with any of the following:

  * History of myocardial infarction within six months
  * Patients with corrected QT (QTc) prolongation \> 500 msec or other significant electrocardiogram (ECG) abnormality noted within 14 days of treatment

    * For patients enrolled in the Phase 1-T portion of the protocol, the QTc should not exceed 470 msec
  * New York Heart Association (NYHA) classification of III or IV
  * If cardiac function assessment is clinically indicated or performed: left ventricular ejection fraction (LVEF) less than normal per institutional guidelines, or \< 55%, if threshold for normal not otherwise specified by institutional guidelines
  * Condition requiring concurrent use of drugs or biologics with pro-arrhythmic potential
* History of stroke or transient ischemic attack within six months
* Patients may not have any evidence of pre-existing inadequately controlled hypertension (defined as a systolic blood pressure \[BP\] of \> 140 mmHg or a diastolic BP of \> 90 mmHg), and must have a normal blood pressure (=\< 140/90 mmHg) taken in the clinic setting by a medical professional within 2 weeks prior to starting study; patients with hypertension may be managed with up to a maximum of three antihypertensive medications; patients who are on three antihypertensive medications must be actively followed by a cardiologist or blood pressure specialist for management of blood pressure while on protocol
* Any prior history of hypertensive crisis or hypertensive encephalopathy
* Clinical significant peripheral vascular disease or vascular disease (aortic aneurysm or aortic dissection)
* Unstable angina
* Major surgical procedure, open biopsy, or significant traumatic injury within 28 days prior to starting cediranib
* History of abdominal fistula, gastrointestinal perforation or intra-abdominal abscess
* Current signs and/or symptoms of bowel obstruction or signs and/or symptoms of bowel obstruction within 3 months prior to starting study drugs
* Current dependency on intravenous (IV) hydration or total parenteral nutrition (TPN)
* Evidence of coagulopathy or bleeding diathesis; therapeutic anticoagulation for prior thromboembolic events is permitted
* Uncontrolled intercurrent illness including, but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant women are excluded from this study because cediranib and olaparib are agents with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk of adverse events in nursing infants secondary to treatment of the mother with cediranib and olaparib, breastfeeding should be discontinued if the mother is treated with cediranib or olaparib; these potential risks may also apply to other agents used in this study
* Known human immunodeficiency virus (HIV)-positive individuals are ineligible because of the potential for pharmacokinetic interactions with cediranib or olaparib; in addition, these individuals are at increased risk of lethal infections when treated with marrow-suppressive therapy
* Patients may not use natural herbal products or other "folk remedies" while participating in this study
* No features suggestive of myelodysplastic syndrome (MDS) or acute myelogenous leukemia (AML) on peripheral blood smear or bone marrow biopsy, if clinically indicated