The senator was referring to Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellSenate passes 0B defense bill Overnight Health Care: New GOP ObamaCare repeal bill gains momentum Overnight Finance: CBO to release limited analysis of ObamaCare repeal bill | DOJ investigates Equifax stock sales | House weighs tougher rules for banks dealing with North Korea MORE's (R-Ky.) Thursday invitation to Obama to come to the Capitol "any time this afternoon if he’s available" to meet with Republicans. The president is scheduled to be in Philadelphia Thursday afternoon for fundraisers. 

Cornyn was one of several Republican senators who came to the floor to express their disappointment in Obama, after the president scolded Republicans in his Wednesday press conference for resisting tax increases in a deal to raise the debt ceiling. The president also mocked Congress's work schedule.

“They're in one week, they're out one week," he said. "And then they're saying, ‘Obama has got to step in. You need to be here.’ I’ve been here. I’ve been doing Afghanistan and bin Laden and the Greek crisis. You stay here. Let’s get it done.”

Those remarks prompted McConnell's invitation and the subsequent Republican criticism.

"He himself has displayed an astounding lack of leadership," Cornyn said of the president.

"Frankly he should be embarrassed but, unfortunately, the threshold for embarrassment here in Washington seems to be much higher here ... than in the rest of the country,” Cornyn said. 

“It rankles many of us to have the president engage in such blatant demagoguery and blame-shifting when he himself has failed to take responsibility for his duties," he added.