There were very, very few people Tony - and by extension, JARVIS - would take a call from; even fewer he'd drop everything and jump into the armor and blast off for a fast flight south - and still fewer who weren't a pretty lady, to that.
When it was Bruce? He dropped everything, armored up, and casually broke the sound barrier on the way south to Gotham. A soon as he got in range, he queried the Cave's security, and sighed a little when it refused to open. So Bruce had locked everything down. Alfred had told him that much. And now he had a nice smack in the face to get it open again, because he was worried about what would happen if Bruce decided to lock himself away...and curious as to why.
It still took him a long hour, hovering over the cave, for JARVIS and his combined efforts to crack the seal to let him fly through the waterfall and into the Cave without being subjected to a few nasty EMP pulses and electrical jolts, to say nothing of the rest of the automated defenses.
Through the waterfall, flicking on the night vision, then autopilot to let JARVIS look ahead and calculate best angles and speeds, through the passage with far too many stalactites and stalagmites to let him, much less the Batmobile through - really, Bruce, how did this work? Then he was through, and into the vast expanse of the cave.
He was tried, worried, and needed a drink. Possibly in that order. He kicked the repulsors into reverse, and slowly glided down to a convenient open space. "Three Screwdrivers. And then two actual screwdrivers to get me out of this," he called over. If he knew Bruce, that should get an irritated grunt, if that. He clumped down the landing pad and to the computers. "I'll take a side of 'what the fuck is going on'," he added.