Well to start with you're capturing a single number, by adding stash around the EE channels tags, it will only ever capture/hold the last number or 0.

To hold all numbers you should use something like the following (**NOTE:** This is a theory and not verbatim code, refer the documents and experiment) I've done similar but not with Favorites so cannot confirm code of usage:

    {exp:channel:entries channel="topic-content" search:topic-title="advertising" category="5" search:topic-view="IS_EMPTY" dynamic="no" parse="inward"}  
        {exp:stash:append name="fav_count" scope="user" trim="yes"}{exp:favorites:count entry_id="{entry_id}" favorites_member_id="CURRENT_USER"}+{/exp:stash:append}
      {/exp:channel:entries}
    //NEW - Add bogus 0 to end of stash
    {exp:stash:append name="fav_count" scope="user" trim="yes"}0{/exp:stash:append}

What this does is append the result of the favourites tags, trims any whitespace which would kill the sum action (next), adds a scope to the current user as well as adding the "+" symbol, which we can now utlise like this:

    {exp:stash:parse}
      {exp:mx_calc parse="inward" expression="sum({exp:stash:get name='fav_count' scope='user'})"}
    {/exp:stash:parse}

Assuming the stash capture results look like this:
`0+1+2+0+` the sum would render like exp:mx_calc parse="inward" expression="sum(0+1+2+0+)

Failing that (or variations). Wrap each result in a unique class and use Javascript to find, record, add up and inject into a result tag.

**UPDATE:**
Added a trailing '0' to the original capture (the trailing '+' was borking the math module.
Inside the math function, used single quotes rather then doubles.


WORKING CODE!

{exp:channel:entries channel="topic-content" search:topic-title="advertising" category="5" search:topic-view="IS_EMPTY" dynamic="no" parse="inward" }  
{exp:stash:append name="fav_count" parse_tags="yes" trim="yes" parse_conditionals="yes"}
{exp:favorites:count entry_id="{entry_id}" favorites_member_id="CURRENT_USER"}
{if "{count}" != "{total_results}"}+{/if}
{/exp:stash:append}
{/exp:channel:entries}

{exp:stash:parse process="end"}
{exp:mx_calc expression="sum({stash:fav_count})"} 
{/exp:stash:parse}