Administrative Services Agreement between Leo Holdings III Corp and Leo Investors III LP

Exhibit 10.5

Leo Holdings Corp. II

Albany Financial Center

South Ocean Blvd

Suite #507

P.O. Box SP-63158

New Providence, Nassau,

The Bahamas

[        ], 2021

Leo Investors III LP

Albany Financial Center

South Ocean Blvd

Suite #507

P.O. Box SP-63158

New Providence, Nassau,

The Bahamas

Ladies and Gentlemen:

This letter will confirm our agreement that, commencing on the effective date (the “Effective Date”) of the registration statement (the “Registration Statement”) for the initial public offering (the “IPO”) of the securities of Leo Holdings III Corp (the “Company”) and continuing until the earlier of (i) the consummation by the Company of an initial business combination or (ii) the Company’s liquidation (in each case as described in the Registration Statement) (such earlier date hereinafter referred to as the “Termination Date”), Leo Investors III LP Limited Partnership shall take steps directly or indirectly to make available to the Company certain office space, secretarial and administrative services as may be required by the Company from time to time, situated at 100 Wilshire Boulevard, Los Angeles, CA 90401 (or any successor location). In exchange therefore, the Company shall pay Leo Investors III LP or one of its affiliates a sum of $10,000 per month, respectively, on the Effective Date and continuing monthly thereafter until the Termination Date. Leo Investors II Limited Partnership hereby agrees that it does not have any right, title, interest or claim of any kind in or to any monies that may be set aside in a trust account (the “Trust Account”) that may be established upon the consummation of the IPO (the “Claim”) and hereby waives any Claim it may have in the future as a result of, or arising out of, any negotiations, contracts or agreements with the Company and will not seek recourse against the Trust Account for any reason whatsoever.

Signature Page to Administrative Services Agreement