You can find this in Villani's "small book", theorem Theorem 8.13 in [Villani, C. (2003). Topics in optimal transportation (Vol. 58). American Mathematical Soc.]
I can also recommend looking at Filippo Santanbrogio's more recent and applied book [Santambrogio, F. (2015). Optimal transport for applied mathematicians. Birkäuser, NY, 55(58-63), 94.] in particular chapter 7 (a more general version of your statement can be found in section 7.2.2)