On Tuesday, Sen. John Cornyn joined other Senate Republicans to block a bill repealing the “Don’t Ask, Don’t Tell’’ policy against gays serving openly in the military.

Tonight, he will receive an award from a gay-rights group.

Sen. John Cornyn (R., Texas) (Photo by Alex Wong/Getty Images)

The Log Cabin Republicans are scheduled to give Cornyn a leadership award, despite his “Don’t Ask’’ vote and his opposition to same-sex marriage.

R. Clarke Cooper, executive director of the Log Cabin Republicans, said the group was looking past those legislative disputes to see political opportunities in helping Republicans rebuild their party for the 2010 midterm elections. Cornyn is chairman of the GOP’s Senate campaign arm, the National Republican Senatorial Committee.

“We are looking for ways to build bridges instead of tearing them down,’’ Cooper said.

He argued that the rise of the fiscally conservative tea party movement could help the cause of gays in the GOP, because the movement tends to be of a libertarian bent…