The HUD displays your best time in race mode too.

I was watching a video with gameplay of Pure in VR and I noticed that above the supposed PSP screen it shows the best time for the track. However, it seems to be 1 screen above the normal one so you can't see it when playing normally.

Perhaps they did that because in was unnecessary for races, but removing it caused issues with the game, so they decided to put it somewhere else.

Here is the video: https://www.youtube.com/watch?v=ZvPnn-TC-B0

Looks like it displays your fastest lap time for that race.

I think it would've been cool if they at least kept that as an option that can be enabled/disabled. It's not as useful in races as it is in time trial, but it would still aid in providing information about your pacing during a given lap (to an extent).