Why did Peter and John go to the temple at the hour of prayer if Christians aren't under the law?
Peter and John went to the temple at the required hour of prayer because they were still under the law and the covenant economy of Israel, not yet understanding the grace of God that would be poured out later. The temple remained central to Israel's religious scene during Acts.
Les explains that Paul never teaches a specific ritual of prayer, yet Peter and John are still going to the temple at the required hour of prayer. That shows they are still under the law and have no concept yet of the grace of God that would be poured out later. They are still in the covenant economy going back to Exodus 19, where Israel, if they accepted their king, could be a kingdom of priests. They went up at about the ninth hour because it was the hour of prayer. After healing the lame man, the people ran together to them at Solomon's porch, which was part of the temple. Les points out the temple was not removed from Israel's religious scene until 69 or 70 A.D.; the crucifixion was about 29 A.D. in his reckoning, with some saying 33. That gives a 40-year period from the crucifixion until the temple's destruction and Israel's dispersion, and the book of Acts covers that whole period. At this point it is just a few weeks after Pentecost.
Now Peter and John went up together into the temple at the hour of prayer, being the ninth hour.
More Questions
Where did Peter and John go in Acts 3, to the church or to the temple?
In Acts 3 Peter and John go up to the temple rather than out to the church, showing the continued temple setting of early Acts.