Exhibit 10.4

KOSMOS ENERGY LTD.

AMENDMENT NUMBER TWO
TO CONSULTING AGREEMENT

THIS AMENDMENT NUMBER TWO TO CONSULTING AGREEMENT (this “Amendment”), effective as of January 1, 2013 (the “Effective Date”), by and between Kosmos Energy Ltd., a company incorporated under the laws of Bermuda (“Kosmos”), and John R. Kemp (“Kemp”). Unless specifically set forth otherwise, reference to the “parties” in this Amendment refers solely to Kosmos and Kemp.

WITNESSETH

WHEREAS, Kosmos and Kemp entered into a Consulting Agreement dated October 31, 2011 and amended effective as of January 1, 2012 (the “Consulting Agreement”) pursuant to which Kemp serves as a consultant to perform such services as Kosmos may reasonably request from time to time during the term of the Consulting Agreement, in addition to his duties as a member of the Kosmos Board of Directors; and

WHEREAS, Kosmos and Kemp desire to amend certain provisions of the Consulting Agreement as set forth in this Amendment.

NOW, THEREFORE, in consideration of the promises and mutual covenants and agreements set forth herein and for other good and valuation consideration, the receipt and sufficiency of which are hereby acknowledged, the parties, intending to be legally bound, agree as follows:

1.             Capitalized Terms.  All capitalized terms used but not defined in this Amendment shall have the same meaning as prescribed in the Consulting Agreement.

2.             Amendment of Section 3.  Section 3 of the Consulting Agreement shall be amended by replacing the first sentence thereof in its entirety with the following:

3. Compensation. As payment for Kemp’s fulfillment of the Consulting Services and covenants set forth in this Agreement, Kosmos shall pay and provide to Kemp: (i) during the period beginning on January 1, 2013 and ending on the expiration or earlier termination of this Agreement, $61,000 per month payable in arrears; and (ii) an award of 12,000 restricted common shares of Kosmos, or a restricted share unit award with respect to 12,000 common shares of Kosmos, as determined by the Compensation Committee of the Board of Directors of Kosmos in its sole discretion, to be granted by Kosmos to Kemp on the first day of each Renewal Term, if any, which award shall become 100% vested (and, in the case of an award of restricted share units, 100% of the common shares underlying such award shall become issuable on or within 30 days after the date of such vesting) at the expiration of each such Renewal Term (collectively, the “Compensation”).

3.             Amendment of Section 4.  Section 4 of the Consulting Agreement shall be amended by deleting therefrom the phrase “in traveling from Houston, Texas to Dallas, Texas.”

4.             Execution and Delivery.  This Amendment may be executed in several counterparts, all of which will together constitute a single agreement among the parties. Delivery by electronic transmission of an executed counterpart of the signature page to this Amendment shall be as effective as delivery of a manually executed counterpart of this Amendment.

5.             No Other Amendments.  Except as modified by this Amendment, all provisions of the Consulting Agreement shall continue in full force and effect.

[Signature Page Follows]

IN WITNESS WHEREOF, the parties hereto have executed this Amendment or have caused this Amendment to be executed by their duly authorized representatives to be effective as of the Effective Date.

KOSMOS ENERGY LTD.
By:	/s/ W. Greg Dunlevy
	Name: W. Greg Dunlevy
	Title:   Executive Vice President and Chief Financial Officer
	/s/ John R. Kemp
	JOHN R. KEMP