I have brewed tea outdoors when the weather was over 100 degrees F. The brewer would even receive some afternoon sun on occasion. When i checked the microbe population with a microscope they were still doing just fine. But the heat changed my ingredients and brew time because of the different oxygen levels.
The main issue i can see with a warmer brew temp is hotter
water does not have as high a capacity to hold dissolved oxygen. Depending on the timing of your temperature fluxuations, say your initial growth explosion of microbes happened in 6 hours at starting temp. If the water heats up at that point the o2 level will drop and the fluid will no longer be able to keep your microherd alive. Use your nose on this, by taking nice long sniffs over the brew. Once most things are dead and the o2 is too low for aerobic microbes the stench will set in, a good brew pretty much never smells funky or sour. Once it dies it will only take a matter of hours to smell bad.
I would guess the heat is coming from the pump and not the microbe action, with an ambient temp of 60 i would expect a brew to be closer to that temp give or take a little, not 30 degrees off.
A pump that creates heat in the water is not ideal, but you can adjust your levels and still brew some good stuff. If you add less compost and less fuel than you might end up with a perfectly fine brew still, just play with the amounts and keep smelling.