Exhibit (a)(5)(H)

Inspire Brands and Dunkin’ Brands Announce Waiver of Inside Date Condition in Connection with Proposed Transaction

Atlanta, GA and Canton, MA – December 7, 2020 – Inspire Brands, Inc. (“Parent”) and Dunkin’ Brands Group, Inc. (“Dunkin’ Brands”) (NASDAQ: DNKN), announced today that Parent and Vale Merger Sub, Inc. (“Purchaser”) have waived the “Inside Date Condition” as described in the Offer to Purchase in connection with Parent’s previously-announced tender offer to acquire all of the outstanding shares of common stock of Dunkin’ Brands at a price of $106.50 per share in cash, effectuated by Purchaser, a wholly-owned indirect subsidiary of Parent. The tender offer is scheduled to expire at one minute following 11:59 p.m., Eastern Time, on Monday, December 14, 2020, unless further extended or earlier terminated in accordance with the terms of the merger agreement. All other terms and conditions of the tender offer remain unchanged.

The tender offer is being made in connection with the Agreement and Plan of Merger, dated as of October 30, 2020 (the “Merger Agreement”), by and among Dunkin’ Brands, Parent and Purchaser. Pursuant to the Merger Agreement, following the consummation of the Offer and the satisfaction or waiver of all conditions, Purchaser will merge with and into Dunkin’ Brands and will cease to exist, with Dunkin’ Brands continuing as the surviving corporation. All outstanding shares of Dunkin’ Brands’ common stock, other than shares (i) irrevocably accepted for purchase by Purchaser in the offer, (ii) held in treasury by Dunkin’ Brands or owned by any direct or indirect wholly-owned subsidiary of Dunkin’ Brands, (iii) owned by Parent or Purchaser or any direct or indirect wholly-owned subsidiary of Parent or (iv) that are entitled to demand appraisal and for which appraisal rights have been properly demanded in accordance with the Delaware General Corporation Law, will be automatically cancelled and converted into the right to receive cash equal to the $106.50 offer price per share.

About Inspire Brands

Inspire Brands is a multi-brand restaurant company whose current portfolio includes more than 11,000 Arby’s, Buffalo Wild Wings, SONIC Drive-In, Rusty Taco, and Jimmy John’s restaurants worldwide. The company was founded in 2018 and is headquartered in Atlanta, Georgia. Inspire is majority-owned by affiliates of Roark Capital Group. For more information, visit InspireBrands.com.

About Dunkin’ Brands Group, Inc.

With more than 20,000 points of distribution in more than 60 countries worldwide, Dunkin’ Brands Group, Inc. (Nasdaq: DNKN) is one of the world’s leading franchisors of quick service restaurants (QSR) serving hot and cold coffee and baked goods, as well as hard-serve ice cream. At the end of the third quarter of fiscal year 2020, Dunkin’ Brands’ 100 percent franchised business model included over 12,500 Dunkin’ restaurants and almost 8,000 Baskin-Robbins restaurants. Dunkin’ Brands Group, Inc. is headquartered in Canton, Mass.

Important Information

This document is for informational purposes only and is neither an offer to purchase nor a solicitation of an offer to sell any shares of common stock of Dunkin’ Brands or any other securities. Purchaser, a direct, wholly owned subsidiary of Parent, has filed a tender offer statement on Schedule TO, including an offer to purchase, a letter of transmittal, and related documents with the SEC and Dunkin’ Brands has filed a solicitation/recommendation statement on Schedule 14D-9 with the SEC with respect to the tender offer. The offer to purchase shares of Dunkin’ Brands’ common stock is only being made pursuant to the offer to purchase, the

letter of transmittal and related documents filed as a part of the Schedule TO, in each case as amended from time to time. THE TENDER OFFER MATERIALS (INCLUDING THE OFFER TO PURCHASE, THE RELATED LETTER OF TRANSMITTAL AND CERTAIN OTHER TENDER OFFER DOCUMENTS) AND THE SOLICITATION/RECOMMENDATION STATEMENT ON SCHEDULE 14D-9 CONTAIN IMPORTANT INFORMATION. SHAREHOLDERS OF DUNKIN’ BRANDS ARE URGED TO READ THESE DOCUMENTS, AS FILED AND AS MAY BE AMENDED FROM TIME TO TIME, CAREFULLY BECAUSE THEY CONTAIN IMPORTANT INFORMATION THAT SUCH HOLDERS SHOULD CONSIDER BEFORE MAKING ANY DECISION REGARDING TENDERING THEIR SHARES. Investors and security holders may obtain a free copy of these statements and other documents filed with the SEC at the website maintained by the SEC at www.sec.gov. In addition, free copies of these documents may be obtained by contacting Innisfree M&A Incorporated, the information agent for the tender offer, toll free at (888) 750-5834 (for shareholders), or collect at (212) 750-5833 (for banks and brokers).

CAUTIONARY NOTE REGARDING FORWARD-LOOKING STATEMENTS

This release contains forward-looking statements and projections within the meaning of Section 27A of the Securities Act of 1933 and Section 21E of the Securities Exchange Act of 1934. Generally, these statements may be identified by the use of words such as “expect,” “intend,” “anticipate,” “believe,” “estimate,” “potential,” “should” or similar words and include, among other things, statements about the potential benefits of the proposed transaction, the prospective performance and outlook of the surviving company’s business, performance and opportunities, the ability of the parties to complete the proposed transaction and the expected timing of completion of the proposed transaction. Forward-looking statements are based on management’s current expectations and beliefs, as well as a number of assumptions, estimates and projections concerning future events and do not constitute guarantees of future performance. These statements are subject to risks, uncertainties, changes in circumstances, assumptions and other important factors, many of which are outside management’s control, that could cause actual results to differ materially from the results discussed in the forward-looking statements. In particular, some of the factors that could cause actual future results to differ materially from those expressed in any forward-looking statements include, among others: (i) uncertainties as to the timing and expected financing of the tender offer; (ii) the risk that the proposed transaction may not be completed in a timely manner or at all; (iii) the possibility that competing offers or acquisition proposals for the Dunkin’ Brands will be made; (iv) uncertainty surrounding how many of Dunkin’ Brands’ stockholders will tender their shares in the tender offer; (v) the possibility that any or all of the various conditions to the consummation of the tender offer may not be satisfied or waived, including the failure to receive any required regulatory approvals from any applicable governmental entities; (vi) the possibility of business disruptions due to transaction-related uncertainty; (vii) the occurrence of any event, change or other circumstance that could give rise to the termination of the merger agreement; (viii) the risk that stockholder litigation in connection with the proposed transaction may result in significant costs of defense, indemnification and liability; (ix) Parent’s ability to realize the synergies contemplated by the proposed transaction and integrate the business of the company; (x) Parent’s level of leverage and debt, including covenants that restrict the operation of its business; (xi) Parent’s ability to service outstanding debt or obtain additional financing; and (xii) other factors as set forth from time to time in Dunkin’ Brands’ filings with the SEC, including its Annual Reports on Form 10-K and Quarterly Reports on Form 10-Q, as well as the tender offer statement, solicitation/recommendation statement and other tender offer documents that will be filed by Parent and Dunkin’ Brands, as applicable. Therefore, you should not place undue reliance on such forward-looking statements. All forward-looking statements are based on information available to management on the date of this communication, and we assume no

obligation to, and expressly disclaim any obligation to, update or revise any forward-looking statements, whether as a result of new information, future events or otherwise. You are cautioned not to place undue reliance on these forward-looking statements, which speak only as of the date hereof.

Media Contacts

For Inspire Brands:

Christopher Fuller, Chief Communications Officer

Press@InspireBrands.com

For Dunkin’ Brands:

Karen Raskopf, Chief Communications and Sustainability Officer

Press@DunkinBrands.com

781-737-5200

Investor Contacts

For Dunkin’ Brands:

Stacey Caravella, Senior Director – Investor Relations

Investor.Relations@DunkinBrands.com

781-737-3200