The fate of boundary terms in dimensional reductions

Gibbons-Hawking-York (GHY) terms are typically neglected when performing dimensional reductions of gravitational theories. We consider the reduction of such terms for both two-derivative and four-derivative theories in general dimensions. We demonstrate a robust consistency wherein the GHY term in the original, higher-dimensional theory translates directly to the appropriate GHY term in the dimensionally reduced theory. In particular, this gives a novel way of generating such terms for higher-derivative corrections. We carry out this procedure for Gauss-Bonnet, Chern-Simons modified, and $f(R)$ gravities to derive novel boundary terms.