"Mr. Hawley’s challenge is not unprecedented... Democrats in both the House and Senate challenged certification of the 2004 election results..."

"... and House Democrats tried on their own to challenge the 2016 and 2000 outcomes, though without Senate support. ... Senator Barbara Boxer of California... briefly delayed the certification of George W. Bush’s victory... cit[ing] claims that Ohio election officials had improperly purged voter rolls... which Mr. Bush carried by fewer than 120,000 votes. Nancy Pelosi, then the House Democratic leader, supported the challenge.... The House voted 267 to 31 against the challenge and the Senate rejected it 74 to 1...  After the 2016 election, several House Democrats tried again, rising during the joint session to register challenges against Mr. Trump’s victory over Hillary Clinton in various states. The Democrats cited reasons ranging from long lines at polling sites to the Kremlin’s election influence operation."

From "Hawley Answers Trump’s Call for Election Challenge/The Republican senator said he would object to certifying the Electoral College votes on Jan. 6, a move that is unlikely to alter President-elect Joseph R. Biden Jr.’s victory" (NYT). 

So... in the last three decades, every time a Republican won, Congressional Democrats challenged the certification of the election, and every time a Democrat won, Congressional Republicans did not challenge the certification.

That certainly puts a different light on what Josh Hawley is doing!

Either challenging the certification is the norm or it is not. It can't be the norm for Democrats and abnormal when a Republican does the same thing. Either Congress has a role in looking into the workings of the state elections or it does not. It can't be that the role is to question Republican victories and rubber-stamp Democratic victories.

I can see — in the NYT write up — the basis for arguing that there actually should be a lopsided role. To fill out something I elided above: "In challenging those results Democrats cited claims that Ohio election officials had improperly purged voter rolls and otherwise disenfranchised hundreds of thousands of voters in the state...." 

The argument that's hinted at is that there should be heightened scrutiny where the challenge has to do with discrimination against a traditionally discriminated against group.