I cannot take my phone into work. Maybe I should take just the battery in...
This would be an option if you've to leave your phone on your bike while working hours.
An other option would be to measure the battery's temperature (while phone's on stand-by) on your bike's storage compartment, leaving the bike on a parking place.
A battery monitor app like Battery Monitor Widget would show you the battery's temperature on a log or as graph by time.
Depending on the measured temperature you could decide your next step
... but will it catch fire?
My HTC phone's safety and regulatory guide says:
"...
Handle battery packs carefully
This product contains a Lithium-ion polymer or Lithium-ion battery. There is a risk of fire and burns if the battery pack is handled improperly. Do not attempt to open or service the battery pack. Do not disassemble, crush, puncture, short external contacts or circuits, dispose of in fire or water, or expose a battery pack to temperatures higher than 60˚C (140˚F).
WARNING: To reduce risk of fire or burns, do not disassemble, crush, puncture, short external contacts, expose to temperature above 60