Q. Is there any historical evidence that Apostle Paul was boiled in hot oil?
The Bible doesn’t tell us about the means or circumstances of Paul’s death. But it does preserve this statement in his second letter to Timothy: “For I am already being poured out like a drink offering, and the time for my departure is near.” Most interpreters understand this to mean that Paul expected to be executed for his faith at the conclusion of his trial in Rome under emperor Nero in the AD 60s.
Some traditional accounts provide further details. The Acts of Paul, an apocryphal work written around the middle of the second century, says that Nero condemned him to death by beheading. A lively legend makes this detail seem accurate: One ancient story about why a certain location in Rome is called the “Three Fountains” is that when Paul was beheaded, his head bounced on the ground three times and a fountain sprang up from each spot. Though the story is fanciful, it would probably never have gotten into circulation if it were known that Paul had been executed some other way, and so it suggests that Paul indeed was beheaded. We can have greater confidence in the work of Eusebius, a very careful researcher, who wrote early in the fourth century in his Ecclesiastical History, “It is … recorded that Paul was beheaded in Rome itself … during Nero’s reign.”
There is a tradition that associates a different apostle with boiling in oil, however. Likely around the end of the second century, Tertullian wrote in The Prescription of Heretics that the apostle John was thrown into a cauldron of boiling oil in the Colosseum, but he suffered no ill effects from what would otherwise have been a gruesome method of execution, and so his sentence was commuted to banishment. We cannot corroborate Tertullian’s report. But it does show, along with the accounts of the sufferings and executions of the other apostles, that the first followers of Jesus stayed loyal to him right to the death, even if this meant enduring the worst tortures that the Romans might inflict.
Good Question 
What evidence do you provide as proof that’s the Acts of the apostles is apocrypha and that it was written at the time you claim?
Scholars have examined the text which indicate that they were written between 60 and 70 CE, which makes this claim incorrect.
Additionally here is some information about the probable cause of Paul’s death. While it cannot be proved the following provides a very good reason why Paul would have been beheaded.
The fact that he was a Roman citizen would exempt him, under Roman laws, from death by lingering torture, in the forms in which it was inflicted on many of his Christian brethren. It would save him from the ignominy of crucifixion, and would thus distinguish his death from that of Peter, who had no claims to Roman citizenship, and who, wherever he died, was probably put to death, like his Master, on a cross (comp. John xxi. 18). There were two modes of beheading among the Romans:—the one by the lictor’s ax; the other by military execution with the sword. In the former case, the criminal was tied to a stake, scourged with rods, and then beheaded;1 in the latter case, the executioner was commonly one of the Imperial bodyguards, and the execution was performed in presence of a centurion, whose duty it was to see the sentence carried out. It is every way probable that Paul was executed in this latter mode.
This post does not refer to the Acts of the Apostles, which all Christians accept as canonical and as inspired Scripture. Rather, it refers to a different work known as the “Acts of Paul.” Besides the stories about the apostles recorded in the biblical book, many others circulated and were collected and written down. These have not been judged by the community of Jesus’ followers to be authentic and authoritative, but they at least provide evidence of what was believed about the apostles in the early centuries. The “Acts of Paul” is one such collection, and I cite it as early evidence that Paul was understood to have been beheaded by Nero. Thank you for the further information you provided about Roman execution.