Long ago, at many times and in many ways, God spoke to our fathers by the prophets, but in these last days he has spoken to us by his Son, whom he appointed the heir of all things, through whom also he created the world. (Hebrews 1:1-2) Did the author believe he was living in the last days?

Not necessarily. Notice how it says that in the past God spoke through the prophets, but now in these last days he has spoken through his son. The word ‘last’ could be referring to the span of time between the beginning of Jesus’ ministry and the time of this book’s writing. Possibly being better translated as ‘these recent days.’

But even if that is not the case and the author was implying that he believed he was living in the last days, there would be nothing wrong with this. The parable below is of the Ten Virgins from Matthew 25, which explains why the author would believe this. In fact, it is very likely that most if not all the early Christians believed they were living in the last days.

Then the kingdom of heaven will be like ten virgins who took their lamps and went to meet the bridegroom. Five of them were foolish, and five were wise. For when the foolish took their lamps, they took no oil with them, but the wise took flasks of oil with their lamps. As the bridegroom was delayed, they all became drowsy and slept. But at midnight there was a cry, ‘Here is the bridegroom! Come out to meet him.’ Then all those virgins rose and trimmed their lamps. And the foolish said to the wise, ‘Give us some of your oil, for our lamps are going out.’ But the wise answered, saying, ‘Since there will not be enough for us and for you, go rather to the dealers and buy for yourselves.’ And while they were going to buy, the bridegroom came, and those who were ready went in with him to the marriage feast, and the door was shut. Afterward the other virgins came also, saying, ‘Lord, lord, open to us.’But he answered, ‘Truly, I say to you, I do not know you.’ Watch therefore, for you know neither the day nor the hour. (Matthew 24:1-13)

Christ told us to be ready at all times. We are to treat every day as though we are living in the last days. I see nothing wrong with the author’s statement.