DA40 flight instructor with student reported low fuel pressure indications during climb. The flight crew diverted to a nearby airport.

Narrative

On a training mission from ZZZ to ZZZ1; in climb to 8;000 ft we saw low fuel pressure indications. We ran the associated checklist for the indication which included turning on the electric fuel pump for the aircraft. which stabilized the fuel indications. After turning the fuel pump off again to see if the fuel indications would still be stable; fuel pressure low annunciations rang and looking at the other indications a saw significant rise in fuel flow and significant rise of engine EGT indications. Believing this to be indicative of a fuel system issue I requested a return to ZZZ from ZZZ TRACON who offered [priority] services and which I declined. I did request expedient vectors for the RNAV RWY XX Approach into ZZZ which would have been more expedient than the traffic pattern. After being established on the approach and in VMC with visual contact of the airport I canceled IFR with ZZZ [TRACON]. About 3nm from ZZZ I noticed that airport operations were active on the runway and they advised over the radio to go around. Throughout the go-around procedure I could see that there had been a runway excursion. Airport operations soon after declared that the runway is closed. After consulting with with my flight school i chose to divert to ZZZ2. After noticing the number of aircraft in the area and believing that any further delay could make my fuel situation worse I requested assistance with ZZZ2 Tower to get to the ground as soon as possible. we made straight in for Runway YY and were cleared to land without further issue.