What's the mixer output if LO signal is applied but not RF signal?

ideally nothing, cause there should be isolation between IF and LO (espicially in double balanced mixer)
but if there is some signal at the IF output then u should calculate the ratio between output and LO which should give the isolation ,,,(tis is normally done by PXF)

I was thinking on this equation : IF=RF-LO, so if frequency of RF=0, then IF would have the frequency of LO ? can it be possible in a double-balanced topology ?

The LO to IF isolation is frequency dependent, but may be 20-50 dB also for a double balanced topology. The balance is not perfect and stray capacitance between LO port and IF port of the mixer gives a LO leak at this port.

u can think of it like the input to the LO pair is DC+RF if RF is zero then input is just DC which would result in LO at the IF output but in case of the double balanced , idealy u get at one end of the differential output LO+ and LO- which eleminates each other giving good isolation , note that all this assuming ideal case but in practise u will get good isolation but not infinte

I think you will see some LO leakage, harmonics of LO and some level of DC. The amount of LO leakage depends on the mixer LO-IF isolaton as well as the IF port bandwidth. The DC level, assuming the IF port is DC coupled, depends on the LO-RF isolation.

so is this the typical way people measure LO leakage / isolation? thanks!

the common way is to do PSS followed by PXF