South Korean and Chinese scientists worry that a collapse at the mountain could trigger a radiation leak.
The U.S. president said Pyongyang's latest test was an "embarrassment" to China and urged Seoul to abandon its attempts at diplomacy with Kim Jong Un.
Hydrogen bombs have never been used in warfare and can generate force hundreds of times higher than the atomic bombs used at Hiroshima and Nagasaki.
Pyongyang carried out its sixth and most powerful nuclear test on Sunday, causing a 6.3 magnitude earthquake.
This year, the country has carried out 10 missile tests troubling neighboring countries.
Park Geun-hye also said the North is "merciless" and under an "extreme reign of terror."
Experts have dismissed the registered seismic activity as too low a magnitude for a hydrogen bomb.