EX-10.4

EXHIBIT 10.4

Execution Version

SECONDAMENDMENTTOREVENUEINTERESTFINANCINGAGREEMENT

THISSECONDAMENDMENTTOREVENUEINTERESTFINANCINGAGREEMENT(this “Amendment”), dated as ofJune 29, 2026 (the “Amendment Effective Date”), is made among RHYTHM PHARMACEUTICALS, INC., a Delaware corporation (the “Company”), HCR RHYTHM SPV, LLC, as administrative agent (in such capacity, “Investor Representative”) and the Investors signatory hereto.

Company, the Investors and Investor Representatives are parties to a Revenue Interest FinancingAgreement dated as of June 16, 2022 (as amended, restated or modified from time to time, the “Financing Agreement”).Borrower has requested that the Investor Representative and the Investors agree to certain amendments to the Financing Agreement.The Investor Representative and the Investors have agreed to such request, subject to the terms and conditions hereof.

Accordingly, the parties hereto agree as follows:

SECTION1Definitions;Interpretation.

(a)Terms Defined in Loan Agreement.All capitalized terms used in this Amendment (including in the recitals hereof) and not otherwise defined herein shall have the meanings assigned to them in the Financing Agreement.

(b)Interpretation.The rules of interpretation set forth in Section 1.2 of the Financing Agreementshall be applicable to this Amendment and are incorporated herein by this reference.

SECTION 2    Amendment to the Loan Agreement.

(a)Section7.7(m)-(n)oftheFinancingAgreementisherebyamendedandrestatedinitsentirety(including by adding the new clause (o) below) as follows:

“(m) to the extent constituting Restricted Payments, the Company and its Subsidiaries may make payments of contingent liabilities in respect of any adjustment of purchase price, earn outs, deferred compensation and similar obligations of the Company and its Subsidiaries;

(n) prior to the Minimum Return Date, the Company and its Subsidiaries may make any other Restricted Payments in an aggregate amount not to exceed $1,000,000; and

(o) so long as no Default or Event of Default has occurred or is continuing, the Company may make Restricted Payments in the form of cash dividends on the Company’s outstanding Series A Convertible Preferred Stock in accordance with the terms thereof pursuant to the Amended and Restated Certificate of Designations for the Series A Convertible Preferred Stock in an aggregate amount from and after June 30, 2026 not to exceed $21,000,000.”

SECTION3    Miscellaneous.

(a)References Within Financing Agreement.Each reference in the Financing Agreement to “this Agreement” and the words “hereof,” “herein,” “hereunder,” or words of like import, shall mean and be a referenceto the Financing Agreement as amended by this Amendment.

(b)Expenses.On or prior to the First Amendment Effective Date, the Company shall pay all reasonable and documented fees, charges and disbursements of counsel to the Investor Representative and all reasonable and documented due diligence expenses of the Investor Representative.

(c)Binding Effect.This Amendment binds and is for the benefit of the successors and permitted assigns of each party.

(d)GoverningLaw.THISAMENDMENTANDTHERIGHTSAND OBLIGATIONS OFTHE PARTIES HEREUNDER SHALL IN ALL RESPECTS BE GOVERNED BY AND CONSTRUED IN ACCORDANCE WITH, THE INTERNAL LAWS OF THE STATE OF NEW YORK.

(e)Severability of Provisions.Each provision of this Amendment is severable from every other provision in determining the enforceability of any provision.

(f)Counterparts.This Amendment may be executed in any number of counterparts and by different parties on separate counterparts, each of which, when executed and delivered, is an original, and all taken together, constitute one Amendment.Delivery of an executed counterpart of a signature page of this Amendment by facsimile, portable document format (.pdf) or other electronic transmission will be as effective as delivery of a manually executed counterpart hereof.

(g)Transaction Documents.This Amendment and the documents related thereto shall constitute Transaction Documents.

(h)ElectronicExecutionofCertainOtherDocuments.Thewords“execution,”“execute”, “signed,” “signature,” and words of like import in or related to any document to be signed in connection with this Agreement and the transactions contemplated hereby (including without limitation assignments, assumptions, amendments, waivers and consents) shall be deemed to include electronic signatures, the electronic matching of assignment terms and contract formations on electronic platforms approved by the Investor Representative, or the keeping of records in electronic form, each of which shall be of the same legal effect, validity or enforceability as a manually executed signature or the use of a paper-based recordkeeping system, as the case may be, to the extent and as providedforinanyapplicable law, includingtheFederalElectronic Signatures inGlobal andNational Commerce Act, the New York State Electronic Signatures and Records Act, or any other similar state laws based on the Uniform Electronic Transactions Act.

[BalanceofPageIntentionallyLeftBlank;SignaturePagesFollow]

INWITNESSWHEREOF,thepartiesheretohavedulyexecutedthisAmendment,asofthedatefirst above written.

BORROWER:    RHYTHMPHARMACEUTICALS,INC.,

aDelawarecorporation

By: /s/ Hunter C. Smith    

Name: Hunter C. Smith

Title:ChiefFinancialOfficer

Acknowledgedandagreed:

INVESTORREPRESENTATIVE:    HCRRHYTHMSPV,LLC

By: /s/ Clarke B. Futch    

Name: Clarke B. Futch

Title:AuthorizedPerson

INVESTORS:    HEALTHCAREROYALTYPARTNERSIV,L.P.,

aDelawarelimitedpartnership

By:HealthCareRoyaltyGPIV,LLC,itsgeneralpartner

By: /s/ Clarke B. Futch    

Name: Clarke B. Futch

Title:ChiefExecutiveOfficer

HCRXINVESTMENTSHOLDCO,L.P.,

aDelawarelimitedpartnership

By:HCRXMasterGP,LLC,itsgeneralpartner

By: /s/ Clarke B. Futch    

Name:Clarke B. Futch

Title:    ChiefExecutiveOfficer

HCRPOVERFLOWFUND,L.P.,

aDelawarelimitedpartnership

By:HCRPOverflowGP,LLC,itsgeneralpartner

By: /s/ Clarke B. Futch    

Name:Clarke B. Futch

Title:    ChiefExecutiveOfficer

HCRCANARYFUND,L.P.,

aDelawarelimitedpartnership

By:HCRCanaryFundGP,LLC,itsgeneral partner

By: /s/ Clarke B. Futch    

Name:    Clarke B. Futch

Title:    ChiefExecutiveOfficer

HCRMOLAGFUND,L.P.,

aDelawarelimitedpartnership

By:HCRMolagFundGP,LLC,itsgeneral partner

By: /s/ Clarke B. Futch    

Name:Clarke B. Futch

Title:    ChiefExecutiveOfficer