It can cost more but very likely will not. My own country is an example of that.A pure figurehead HoS is not necessarily cheaper compared to a head of government HoS. It can be the case. A elected HoS will be out of job at one point. In many cases they receive money for security and often a small office and some staff for the rest of their life. Not to forget an amazing pension for a couple of years work. So changing them every 5-10 years still works up a pretty high numbers of people you have to pay for. It’s just a little not so well known fact. Again not true everywhere, but in the end a monarchy is not more expensive than an elected HoS in general. There are good arguments for both models, but money just doesn’t cut it.
I have yet to see the country where money was freed up for worthwhile causes or saved in the individual purse after a monarchy was abolished.
The real savings with getting rid of the monarchy would be the crown estate and all resultant revenue reverting to the state.
Also, re: another post higher up, what does knowing the names of another's country president have to do with how well they are governed? Surely, 'brand recognition' is the least significant concern when one thinks of a fair, just, dedicated and efficient leader?