(Recasts with annual rate) MEXICO CITY, Aug 23 (Reuters) - Mexican inflation cooled in early August compared to the prior month, official data showed on Thursday, backing expectations for steady interest rates in the coming months. Consumer prices in the 12 months through early August rose 4.81 percent, the national statistics agency said. That was above estimates for a 4.74 percent increase but down from a 4.85 percent rate in early July. The agency introduced a new methodology with this data that includes an updated basket of goods and a wider sample of data from more small cities. Mexico's central bank early this month held its benchmark interest rate at a more than nine-year high of 7.75 percent and trimmed its growth forecast. Most economists expect the central bank to start cutting rates next year, according to a poll by Banamex this week. Thursday's data showed the closely-watched annual core index rose 3.60 percent, in-line with expectations. Mexican consumer prices rose 0.34 percent during the first half of August. The core price index , which strips out some volatile food and energy prices, climbed 0.18 percent in early August. (Reporting by Michael O'Boyle Editing by Nick Zieminski)