A warning leaflet, posted at many resting places, has this to say:

Cougar Country

Not many hikers can say they have seen mountain lions in the wild, but that is a possibility in Bryce Canyon's back country.  A cougar attack has never happened here in Bryce Canyon National Park, but if you should have the privilege of encountering a cougar, follow the steps listed below to insure your encounter is a pleasant one:

CHILDREN should be picked up and held until the encounter is over.  ALWAYS keep small children with you; do not let them run ahead.  The majority of cougar attacks are on small children.

STOP! Do not run! Cougars tend to chase things that run.  They associate running with prey. Hold your ground, or back away slowly.  Look the cougar in the eye, and never turn your back.

APPEAR as large as possible.  Stand tall, raise your hands over your head.  Don't try to hide.  Cougars see most people long before the people spot them.

SHOUT, wave your arms.  If the cougar behaves agressively, respond aggressively, throw sticks or rocks, etc.

FIGHT BACK if you are attacked.  Do your best to keep the cougar from getting behind you.  You want to protect the back of your head and neck.  Be as combative as possible.

REPORT any mountain lion sightings at the visitor centre.

When visiting cougar country it is important we understand their behaviour and act accordingly to protect them and ourselves.