I just noticed my highest-level pet now takes 90 minutes to train instead of 60. Is it purely level dependent, or something to do with rating? Anyone know at what points the training time increases, and to how long?

it firstly increases when you reach level 50, and it keeps increasing every 100 levels after that. here're the training times i found on the subeta wiki

Level 0 - 49 : 60 minutes Level 50 - 149: 90 minutes Level 150 - 249: 120 minutes Level 250 - 349: 180 minutes Level 350 - 449: 225 minutes Level 450 - 549: 270 minutes Level 550 - 649: 330 minutes Level 650 - 749: 390 minutes Level 750 - 849: 435 minutes Level 850 - 949: 480 minutes Level 950 - 1049: 525 minutes Level 1050 - 1149: 600 minutes Level 1150 - 1249: 650 minutes Level 1250 - 1349: 700 minutes Level 1350 - 1449: 725 minutes Level 1450 - 1549: 750 minutes Level 1550 - 1649: 800 minutes Level 1650 - 1749: 850 minutes Level 1750 - 1849: 900 minutes Level 1850 - 1949: 950 minutes Level 1950 - 2049: 1000 minutes

I guess there must be a more efficient way to train for really high level pets, because I don't think the site has existed long enough for the top pets to get to where they are by just training...

You can also gain stats by doing battle quests, using boosters, the oracle will give you some random stats if you do her quest when she calls you for one and sometimes your pet will get a radom event where they moan and groan and gain some stats...for sure if it was by training only, pets wouldn't get very far after a certain level, my pet only gets 2 training sessions a day now