Turkey, co-host of COP31, wants a key outcome of the summit to be a commitment by countries to meet 35% of their energy supply from electricity by 2035.
He was the only G7 leader to speak at this year's conference, a sign of faltering support for the COP process that he acknowledged in his address.
The warning matters because it frames Britain's climate stance as the country prepares to take up the G20 presidency next year, a role officials see as a strong indication of where UK priorities will lie.
Burnham opened his speech by talking about the heat Britain experienced last summer. He said"it was not normal". The Met Office said the season was the hottest on record and was made 130 times more likely by climate change.
The summer's effects were concrete. Hundreds of schools were closed, cereal harvests were down 11%, and ambulance services were inundated with calls beyond levels seen during the Covid-19 pandemic.
On cutting the causes of climate change, Burnham said his government remained committed to meeting the country's target of net zero by 2050 while also getting bills down.
"Our children and grandchildren will not judge us by the promises we made, but by what we chose to do,"he said.
Switching petrol and diesel cars to electric vehicles, and moving from gas boilers to electric heat pumps, are key ways to reach that electrification target.
Although there was no royal presence this year, King Charles III sent a message ahead of Burnham's speech.
Environmental campaigners at the summit said the prime minister had to back his warm words with stronger action.
Burnham has also been accused of double standards, though the speech itself restated the net-zero commitment alongside a pledge to bring household bills down.
Other leaders expected to speak on Thursday included island states such as Tonga, Tuvalu and the Solomon Islands. For those nations, the impact of climate change is already at their front door.