London North Eastern Railway (LNER), which runs services on the line, has urged its customers against travel and have offered full refunds on tickets.
In a statement on its website it said 'severe weather' would result in speed restrictions between London King's Cross and Peterborough all day.
The UK could see temperatures soar to a record-breaking 40C, the Met Office said as it warned heatwaves were on the increase as a result of climate change.
"LNER will be running a reduced timetable and we expect our trains to be extremely busy," the statement read.
"We encourage customers to avoid travel on July 25 and where possible travel on either Friday, July 26, Saturday, July 27 or Sunday, July 28. If you choose not to travel, you will be entitled to a full refund."
The scorching temperatures gripping the UK and much of Europe come against a backdrop of global warming of 1C since the Industrial Revolution driven by greenhouse gas emissions.
Higher temperatures are making extreme hot spells more likely and more intense, the experts warn.
A Met Office study into last year's record-breaking summer temperatures found they were about 30 times more likely as a result of climate changecaused by human activities.
Dr Mark McCarthy from the Met Office National Climate Information Centre, said: "Reaching 40C in the UK is an unprecedented event in the observational records of UK climate.
"Historically UK summer heatwaves would typically tend to peak in the low 30s with extreme events reaching the mid-30s."
He said the UK climate had been warming since the middle of the 20th century.
The hottest day of the year has also increased, and in the most recent decade was around 0.8C higher than compared to the period 1961-1990, he said.
"Climate change has increased the likelihood and severity of heatwave episodes across Europe, which will have also increased the risks of a 40C temperature event in the UK."
The Met Office says there is a significant chance of the July maximum temperature record of 36.7C, set at Heathrow in 2015, could be broken on Thursday.
It is also possible the all-time UK temperature record of 38.5C, set in Faversham, Kent, in August 2003, could be broken, and even the 40C threshold could be reached.
Climate projections suggest that by mid century there will be a 50 per cent chance of summers as hot as 2018's heatwave, making the sweltering conditions the norm.
The Government's advisory Committee on Climate Change recently warned the UK was not prepared for a future of more heatwaves, with more action needed to prevent overheating in homes, hospitals and schools, and that even vulnerable people did not consider themselves at risk.
Professor Hannah Cloke, a natural hazards researcher at the University of Reading, said the heatwave served as a reminder of what the country needed to do to prepare for the future.
She said the UK must take "entirely achievable" steps to stop making the problem worse and follow up on commitments to tackle climate change.
But she also said: "Just to cope with the warmer climate that we are already experiencing, we have to make changes to the way we live our lives.
"We need to adapt our homes and buildings to stay cool, change our activities to stay out of harm's way, and take action to protect the most vulnerable people."
And she said authorities such as councils, hospitals, fire brigades and organisations in charge of core infrastructure must act quickly on early warnings to make sure they keep vital services going.