Veliparib, Topotecan Hydrochloride, and Filgrastim or Pegfilgrastim in Treating Patients With Persistent or Recurrent Cervical Cancer

Inclusion Criteria:

* Patients must have persistent or recurrent squamous cell carcinoma, adenosquamous carcinoma, adenocarcinoma or non-squamous cell carcinoma of the cervix with documented disease progression; histological documentation of the original primary tumor is required via the pathology report
* All patients must have measurable disease as defined by RECIST 1.1; measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded); each lesion must be \>= 10 mm when measured by computed tomography (CT), magnetic resonance imaging (MRI) or caliper measurement by clinical exam; or \>= 20 mm when measured by chest x-ray; lymph nodes must be \>= 15 mm in short axis when measured by CT or MRI
* Patient must have at least one "target lesion" to be used to assess response on this protocol as defined by RECIST 1.1; tumors within a previously irradiated field will be designated as "non-target" lesions unless progression is documented or a biopsy is obtained to confirm persistence at least 90 days following completion of radiation therapy
* Patients must not be eligible for a higher priority Gynecologic Oncology Group (GOG) protocol, if one exists; in general, this would refer to any active GOG Phase III protocol or Rare Tumor protocol for the same patient population
* Patients must have a GOG performance status of 0, 1, or 2
* Recovery from effects of recent surgery, radiotherapy, or chemotherapy
* Patients should be free of active infection requiring antibiotics
* Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to registration; continuation of hormone replacement therapy is permitted
* Any other prior therapy directed at the malignant tumor, including chemotherapy, biologic/targeted (non-cytotoxic) therapy and immunologic agents must be discontinued at least three weeks prior to registration; all side effects must have resolved to =\< grade 1 or stabilized, prior to enrolling on this study
* Any prior radiation therapy must be completed at least 4 weeks prior to registration
* Patients MUST have had one prior systemic chemotherapeutic regimen for management of advanced, metastatic, or recurrent squamous or non-squamous cell carcinoma of the cervix; chemotherapy administered concurrent with primary radiation is not counted as a systemic chemotherapy regimen (e.g.; weekly cisplatin); adjuvant chemotherapy given following the completion of radiation therapy (or concurrent chemotherapy and radiation therapy) is not counted as a systemic chemotherapy regimen (e.g.; paclitaxel and carboplatin for up to 4 cycles)
* Patients who are registered during the safety lead-in portion of this protocol are required to have prior pelvic radiation
* Patients must have NOT received more than one previous cytotoxic chemotherapy regimen (either with single or combination cytotoxic drug therapy)
* Patients are allowed to receive, but are not required to receive, biologic/targeted (non-cytotoxic) therapy as part of their one prior systemic chemotherapeutic regimen; patients are allowed to receive, but are not required to receive, biologic/targeted (non-cytotoxic) therapy for management of recurrent or persistent disease
* Patients MUST not be eligible for further curative intent surgical or pelvic radiation treatment for management of recurrent or persistent disease as determined by treating physicians
* Absolute neutrophil count (ANC) greater than or equal to 1,500/mcl
* Platelets greater than or equal to 100,000/mcl
* Creatinine less than or equal to 1.5 x institutional upper limit normal (ULN)
* Bilirubin less than or equal to 1.5 x ULN
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 3 x ULN
* Alkaline phosphatase less than or equal to 2.5 x ULN
* Neuropathy (sensory and motor) less than or equal to grade 1
* Patients must have signed an approved informed consent and authorization permitting release of personal health information
* Patients must meet pre-entry requirements
* 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 and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately
* Patients must have the ability to swallow pills whole

Exclusion Criteria:

* Patients are excluded who have had prior therapy with ABT-888 (veliparib), poly (ADP)-ribose polymerase inhibitors, or topotecan
* Patient with a history of other invasive malignancies, with the exception of non-melanoma skin cancer and other specific malignancies are excluded if there is any evidence of other malignancy being present within the last three years; patients are also excluded if their previous cancer treatment contraindicates this protocol therapy
* Patients who have received prior radiotherapy to any portion of the abdominal cavity or pelvis OTHER THAN for the treatment of cervical cancer within the last three years are excluded; prior radiation for localized cancer of the breast, head and neck, or skin is permitted, provided that it was completed more than three years prior to registration, and the patient remains free of recurrent or metastatic disease
* Patients who have received prior chemotherapy for any abdominal or pelvic tumor OTHER THAN for the treatment of cervical cancer within the last three years are excluded; patients may have received prior adjuvant chemotherapy for cancer, provided that it was completed more than three years prior to registration, and that the patient remains free of recurrent or metastatic disease
* Patients with seizures or history of seizures are ineligible
* Patients with history or evidence upon physical examination of central nervous system (CNS) disease, including primary brain tumor, any CNS metastases, or history of cerebrovascular accident (CVA, stroke), transient ischemic attack (TIA) or subarachnoid hemorrhage within six months of the first date of treatment on this study, are ineligible; patients with CNS metastases must be stable for \> 3 months after treatment and off steroid treatment prior to study enrollment