It has no effect (see updated answer) I don't understand why...

UPDATE:

date_deadline field in MailActivity object (in mail_activity.py file):

date_deadline = fields.Date('Due Date', index=True, required=True, default=fields.Date.context_today)

Log declaration (I don't know if it is declared in some JS file or not):

<li t-if="!_.isEmpty(records)">
        <ul class="nav o_activity_log">
            <t t-foreach="_.keys(records)" t-as="key">
                <t t-set="logs" t-value="records[key]" />
                <li class="o_activity_label">
                    <strong t-attf-class="o_activity_color_#{key}">
                        <t t-esc="selection[key]" /> (<t t-esc="logs.length"/>)
                    </strong>
                </li>
                <li t-foreach="logs" t-as="log" t-att-data-activity-id="log.id" class="o_schedule_activity">
                    <div class="o_activity_title pull-left">
                        <span t-attf-class="fa #{log.icon} fa-fw" />
                        <strong>
                            <t t-esc="log.title_action or log.activity_type_id[1]" />
                        </strong>
                        <div>
                            <span class="fa fa-clock-o fa-fw" />
                            <span t-att-title="log.date_deadline"><t t-esc="log.label_delay" /></span>
                            <t t-if="log.user_id[0] != uid">
                                <span class="ml4 fa fa-user" />
                                <span><t t-esc="log.user_id[1]" /></span>
                            </t>
                        </div>
                    </div>
                    <div class="pull-right">
                        <span class="o_mark_as_done o_activity_link o_activity_link_kanban fa fa-check-circle fa-2x mt4" t-att-data-activity-id="log.id" title="Mark as done" />
                    </div>
                </li>
            </t>
        </ul>
    </li>

Thank you for help !