Even as his bullishly conservative agenda has unsettled the state’s political equilibrium, Mr. Brownback maintains the gregarious, faith-driven manner that has defined him since his days as a United States senator, long-shot presidential candidate and hero of the country’s Christian right.

In the Senate from 1996 to 2011, he vigorously opposed abortion, promoted low taxes and less government, and worked against the genocide in Sudan, which won him praise from some liberals. In Kansas, Mr. Brownback has argued that he is not beholden to ideology. He has, for instance, championed wind energy, supporting the renewable energy production requirement — something conservatives in the state have wanted to eliminate — and he signed legislation to help conserve water in the Ogallala Aquifer.

Kansas has a long history of moderation. It entered the Union as a free state more than a century and a half ago after years of bloody battles against pro-slavery insurgents. The state’s economy — founded on cattle, wheat and other crops — has been balanced more recently by aviation and technology. While the political power currently resides with the far right, this home state of Bob Dole and Dwight D. Eisenhower has a history of being governed from the center. It has elected five Democratic governors over the past 50 years, and many of the Republicans who ran the state called themselves moderates.

This election year, voters will have a chance to decide whether they still feel comfortable with having strayed from a centrist course. Some political analysts say Mr. Brownback is favored to prevail even though his approval ratings have dropped over the years.

Neil Newhouse, a Republican pollster who has worked for Mr. Brownback, said that given President Obama’s roughly 35 percent approval rating in Kansas, it would be difficult for any Democratic challenger to beat an incumbent Republican.

Mr. Brownback’s signature achievement as governor has been eliminating income taxes for almost 200,000 small businesses and deeply cutting individual income taxes. He said that the Kansas economy was already showing signs of moving forward because of the cuts. Unemployment fell to 4.9 percent last December, the lowest level in more than five years and lower than the national average of 6.6 percent in January. When he started his term in January 2011, the state had $876 in the bank; now there is more than $700 million in cash reserves and Kansas personal income, a measure of family earnings, is growing. The state had more than 15,000 new business filings in 2012, a record, and sales tax receipts have risen.