Rituximab, Cladribine, and Temsirolimus in Treating Patients With Newly Diagnosed Mantle Cell Lymphoma

Inclusion Criteria: * Histologically confirmed mantle cell lymphoma (MCL); the diagnosis must be confirmed by NCCTG pre-registration pathology review by Dr. Paul Kurtin or his designate; it is recommended that the biopsy be an excisional biopsy, but adequate core-needle biopsies will be accepted as long as they are considered adequate for registration by Dr. Kurtin or his designate; the tumor must be cyclin D-1 positive by immunohistochemistry or have evidence of a t(11;14) translocation by fluorescence based in situ hybridization (FISH) or cytogenetics * Measurable or assessable disease, defined as at least one of the following: * A lymph node or tumor mass that is \>= 2.0 cm in at least one dimension by positron emission tomography (PET)/computed tomography (CT), CT, magnetic resonance imaging (MRI), or plain radiograph imaging * Splenic enlargement may be used as a measurable parameter if the spleen is palpable \>= 3 cm below the left costal margin * Diffuse infiltration of an organ such as the stomach, bone marrow, peripheral blood, liver, lungs, or bowel by lymphoma without a discrete mass would constitute assessable, but not measurable, disease * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0, 1, 2, or 3 * Life expectancy \>= 12 weeks * Absolute neutrophil count (ANC) \>= 1,500/mm³ * Platelet count (PLT) \>= 100,000/mm³ * Serum creatinine =\< 2.0 mg/dL * Serum total bilirubin (or direct bilirubin if total is abnormal) =\< institutional upper limit of normal (ULN) with or without secondary liver involvement * Serum glutamic oxaloacetic transaminase (SGOT) =\< 3 x institutional ULN (exception: if there is liver involvement, SGOT must be =\< 5 x institutional ULN) * Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * Willingness to return to NCCTG enrolling institution for follow-up * Willingness to provide the blood specimens as required by the protocol * Willingness to provide tissue specimens as required by the protocol * Willing to return to NCCTG enrolling institution for follow-up * Willing to provide blood and tissue specimens as required by the protocol * Willing to abstain from eating grapefruit or drinking grapefruit juice * Willingness to abstain from eating grapefruit or drinking grapefruit juice for the duration of the study Exclusion Criteria * Any prior therapy for mantle cell non-Hodgkin lymphoma including radiation therapy; exception: patient may have undergone a splenectomy for diagnosis, cytopenia, or systematic splenomegaly * Active or uncontrolled infection * Any of the following cardiac conditions: * Uncontrolled high blood pressure * Unstable angina * Active congestive heart failure * Myocardial infarction =\< 6 months * Serious uncontrolled cardiac arrhythmia * Known central nervous system (CNS) involvement * Any of the following: * Pregnant women or women of reproductive ability who are unwilling to use effective contraception while taking the drug and for 12 months after stopping treatment * Nursing women * Men who are unwilling to use a condom (even if they have undergone a prior vasectomy) while having intercourse with any woman, while taking the drug and for 12 months after stopping treatment * Medical or psychiatric conditions which, in the opinion of the investigator, make the patient a poor risk for participation * Known to be human immunodeficiency virus (HIV) positive; HIV testing is not required but should be done if clinically indicated; HIV-positive patients receiving combination anti-retroviral therapy are excluded from the study * Concurrent malignancy =\< 5 years ago; exceptions: carcinoma in situ of the cervix, resected basal cell or squamous cell carcinomas of the skin, or prostate cancer that is in remission following a radical retropubic prostatectomy or radiation therapy; if there is a history of prior malignancy, they must not be receiving other specific treatment (other than hormonal therapy) for their cancer * Known hypersensitivity to rituximab or its components, or to murine proteins * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Prior treatment with an mTOR inhibitor * Autologous or allogeneic stem cell transplant planned as part of initial therapy * Receiving enzyme-inducing antiepileptic drugs (enzyme inducing anti-epileptic drugs \[EIAEDs\]; e.g., phenytoin, fosphenytoin, carbamazepine, oxcarbazepine, phenobarbital, or primidone); any other potent cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inducer such as rifampin, glucocorticoids at greater than adrenal replacement levels, or St. John's wort; or receiving strong CYP3A4 inhibitors \* Note: if these agents are discontinued, temsirolimus therapy can begin \>= 7 days after discontinuation of such agent